Has non-TSI solar output increased over the past century in ways the IPCC climate models ignore?
Probably not.
What Professor Shaviv has done is shown that it is possible to make a simple climate model which replicates climate change by adding certain parameters the IPCC does not use. What they have not shown is whether this has any relation to reality.
The professor is using their own simple model which has not been shown to be connected to reality. Their models include parameters they say the IPCC does not: Indirect Solar Effect (ISE) (which he appears to have made up), Pacific Decadal Oscillation (PDO), and Southern Oscillation Index (SOI) which are variations in ocean temperatures in specific areas. The professor has hand-tweaked their effects to replicate climate change history. They acknowledge their models are simple and loaded with assumptions, which is fine, but there's no indication they have gotten feedback from climate scientists about the usefulness of their model.
Problem is they do not mention this in their article. They speak about the climate with a certainty the models do not offer. All the claims in the article should be prefaced with "if my assumptions are correct" and there's a lot of assumptions.
Does the IPCC Ignore These?
IPCC's Fifth Assessment, the one current as of the article, The Physical Science Basis Chapter 9 assesses oceanic oscillations like the professor uses.

PDO is one of them. They find little or no improvement when adding it to their model.
The IPCC is working on their sixth assessment.
What's new in their model?
Quantifying the role of solar radiative forcing over the 20th century explains...
To sum up, our model accepts the various radiative forcings as knowns, with exception of the new ones (ISE, SOI, and PDO indices) and the Aerosol Indirect Effect (AIE), which we take as a cooling only effect.
Indirect Solar Effect (ISE) and the AA Index
In addition to the standard forcings, we introduce a few more. The first one describes an indirect solar/climate effect (ISE), which may exist in addition to changes in the total solar irradiance (TSI) which we take as well, and which we assume to be proportional to the AA index. We use this geomagnetic index because of several reasons. First, it is an indirect index which can be used to describe the non-thermal activity of the sun. Second, it has a long enough record to cover our simulation time span. And last, we require a proxy which does not have a short time or longtime distortion.... Because of the simplicity of our model, we also implicitly assume that the indirect mechanism which we are looking for (ISE), is proportional to the AA index. In particular, we assume that it has the same phase.

What is the ISE? They don't know.
One should note that the indirect solar effect can be one of several different mechanisms (or a combination of them), such as hypersensitivity of the climate system to UV (Haigh, 1994, Haigh et al., 2010), or climate sensitivity to the atmospheric ionization, governed by solar induced cosmic ray modulation (e.g.,Svensmark, 1998; Tinsley,1990). The present model cannot distinguish between them. It can only address whether such a mechanism, which depends on the non-thermal component of the solar activity, exists.
We also add internal climate forcings – proportional to the Pacific Decadal Oscillation (PDO) index and the Southern Oscillation Index (SOI). In principle, such internal oscillations should arise from an ideal climate model. However, even the standard GCMs do not predict them. We therefore prescribe them “by hand”. This is done because they improve the fit to the observations, which in turn helps to better constrain the other model parameters.
This one is a good example of why the models may not represent reality. The professor does not give a physical reason for including these parameters; they just make the model work better. Admittedly the IPCC does this, too, but in far, far more detail.
National Center for Atmospheric Research (NCAR) describes the PDO as...
The Pacific Decadal Oscillation (PDO) is defined by the leading pattern (EOF) of sea surface temperature (SST) anomalies in the North Pacific basin (typically, polewards of 20°N). The SST anomalies are obtained by removing both the climatological annual cycle and the global-mean SST anomaly from the data at each gridpoint. Positive values of the PDO index correspond with negative SST anomalies in central and western North Pacific (extending eastwards from Japan), and positive SST anomalies in the eastern North Pacific (along the west coast of North America). The positive phase of the PDO is also associated with positive SST anomalies across the central and eastern tropical Pacific. A weak mirror image of these anomalies occur across the South Pacific. Overall, the PDO's spatial pattern resembles that of ENSO. The largest distinction between the PDO and ENSO is their timescales: While ENSO is primarily an interannual phenomenon, the PDO is decadal in scale. Thus, relatively long data records are needed to define and understand the PDO.

NCAR describes the SOI...
The Southern Oscillation Index (SOI) is a time series used to characterize the large scale sea level pressure (SLP) patterns in the tropical Pacific. Monthly mean SLP at Tahiti [T] and Darwin [D] are used. An optimal SOI can be constructed. It consists of [T-D] which is a measure of the large scale phenomena while [T+D] is a measure of small scale and/or transient phenomena that are not part of the large scale Southern Oscillation. The SOI is linked to large scale tropical SST variability and as such is a measure of the "SO" part of the ENSO phenomenon.

The professor goes on to say...
In many previous analyses, the SOI and occasionally the PDO components were removed from the observed SST [Sea Surface Temperature] through different statistical procedures. We chose an alternative, which is to use our own optimization tool, to obtain the SOI and PDO components in the oceanic and land temperature data. Both procedures effectively remove the SOI and PDO components from the observed data set and should produce the same results. Our approach has the disadvantage that it conceptually treats the PDO and SOI as if they are external forcings affecting the climate, clearly this is not the case. The advantage is that it is much simpler to implement.
Prof. Shaviv often chooses simplicity over accuracy. This calls the utility of the model into question.
North Atlantic Oscillation (NAO)
We can try and improve the fit by adding additional “internal oscillations”. The next oscillation expected in relative importance after the SOI and PDO, is the North Atlantic Oscillation (NAO, Hurrel, 1995).
Prof. Shaviv tried adding the NAO, but removed it because "the NAO does not introduce a better fit" and "the ocean–land coupling becomes undetermined once the NAO is included, as it obtains a very wide PDF [Parameter distribution function]". Which is to say, adding the NAO didn't make the model work better, and it might have broken it.
Prof. Shaviv's work
I found some papers by Prof. Shaviv which explain their ideas in more detail. The first one is particularly relevant.
They're all in very reputable journals. I have no reason to doubt them. They appear to provide interesting alternative models of climate change.
The problem is, as near as I can tell, these are all written by physicists. I am not a climate scientist and neither is Prof. Shaviv. They may have found a model that fits the data, but does it fit reality? Their models are loaded with assumptions, which is fine so long as they're clearly documented; which they are. They would have to enlist the help of climate scientists to check if their assumptions are correct and refine their models. They do not appear to have done so, and that's the problem.
The discussion section of Quantifying the role of solar radiative forcing over the 20th century sums it up nicely.
The climate model employed in this analysis is relatively simple. It is an EBM which includes several “boxes”and a diffusive ocean. Nevertheless, it is rich enough to describe many of the basic aspects of the climate, and parameterize them. This allows for an extensive parameter study, something which is nearly impossible with a full global circulation model.
With the model, we have shown that the observed land and ocean anomalies can to a large extent be described as a response to the theoretical radiative forcings... Statistically, the fit improves considerably if we include a solar forcing driver (ISE) other than the variations in the total solar irradiance (TSI).
Decoded: This is a great and affordable simulation. We tweaked the parameters until it matched existing data. We don't know if it relates to reality nor if it has predictive value.
In On climate response to changes in the cosmic ray flux and radiative budget, Prof. Shaviv is again very clear that their model has problems.
Without a detailed physical model for the effects of
cosmic rays on clouds or a detailed enough record of
radiation budget measurements correlated with the solar
cycle, it is hard to accurately determine the quantitative link
between CRF [Cosmic Ray Flux] variations and changes in the global radiation budget. In particular, it is hard to do so without limiting ourselves to various approximations
Decoded: We had to do a bit of clearly documented fudging. Again, fine so long as you document it and remember your model is fudged when you write articles in newspapers.
The solar and Southern Oscillation components in the satellite altimetry data claims to "find that at least 70% of the variance in the annually smoothed detrended altimetry data can be explained as the combined effect of both the solar forcing and the El Niño–Southern Oscillation (ENSO)". However, "assume" appears often in their paper.
Since we do not wish to limit ourselves to a particular mechanism for the solar forcing (such as hypersensitivity to UV or to cosmic rays), we will assume for simplicity that the forcing is harmonic, as we discuss below.
We assume that the mean sea level (MSL) can be described by a long-term linear trend, a harmonic solar contribution, and a term reflecting the ENSO.
Thus, we assume that the sea level can be approximated by...
The above empirical fit assumed a harmonic solar forcing.
This simple model can be improved as is described in Figure A1. First, instead of assuming just a mixed layer...
Again, we assume here that warmer oceans increase the water loss rate.
This assumes the atmosphere keeps a constant relative humidity.
One therefore requires the average temperature of the oceans and the assumption that the oceans heat uniformly.
Again, clearly stating your assumptions is all perfectly normal parts of the scientific process. The problem is when one forgets their model is riddled with unverified assumptions.
The Hiatus That Wasn't.
Prof. Shaviv mentions the "hiatus".
And of course there is the “hiatus.” The IPCC concluded in Chapter 9 of its September 2013 Working Group I report that there had been a 15-year hiatus in Global Surface Meant Temperatures (GSMT) that had not been predicted by a single computer model. Currently, satellite data show that the hiatus has continued over 18 years, even though carbon dioxide has risen significantly. This implies that Earth’s temperature increases less (from the influence of CO2) than IPCC predictions, because those were based on a high climate sensitivity ascribed to CO2.
However, in Possible artifacts of data biases in the recent global surface warming hiatus, June 2015, NOAA argues there was no hiatus. It was due to biases in the data. Once those are removed the hiatus disappears.
Much study has been devoted to the possible causes of an apparent decrease in the upward trend of global surface temperatures since 1998, a phenomenon that has been dubbed the global warming “hiatus.” Here, we present an updated global surface temperature analysis that reveals that global trends are higher than those reported by the Intergovernmental Panel on Climate Change [IPCC], especially in recent decades, and that the central estimate for the rate of warming during the first 15 years of the 21st century is at least as great as the last half of the 20th century. These results do not support the notion of a “slowdown” in the increase of global surface temperature.
It's normal for careful examination of the data by other parties to find biases and provide reinterpretations of the results later. Then everything based on that flawed data has to be reinterpreted. That's science.
Prof. Shaviv's conclusion that CO2 has a lower influence on temperature is based on the "hiatus". It must now be re-examined.
The NOAA paper was published just as Prof. Shaviv was publishing their article in the Financial Post. No dishonesty on their part... unless they continue to make that claim without addressing the NOAA findings. Might be something to ask about.
Models are useful, but they are not reality.
Prov. Shaviv shows one can make a model of climate change using nothing but solar forcing, but do not show that this is reflects reality. To do that, they would need to collaborate with climate scientists who will likely find many factors were omitted from their relatively simple model.
It is useful to make models to explore alternative climate models, this appears to be good science. "Here's one way this could work, we should look into this more" is a fine position. The problem is in the article Prof. Shaviv strays into claiming that it really does work this way.
I found that the sun’s variability as well as unrelated cosmic ray variations, explain a surprisingly large amount of the observed climate variations, from the 11-year solar cycle to geological time scales
"Can explain" has become the definite "explain". All that scientific rigor and honesty in peer-reviewed papers disappears in the article and becomes a declaration that "the Sun raises the seas".
Based on my “galactic view” of climate change, the good news is we’re not doomed. The “carbon risk” of catastrophic global warming or climate change is low. The sun has a far greater, natural influence on climate than many are willing to admit.
Yes, based on their view which may or may not be correct.