TL;DR: It's more complicated than the dichotomy "fat=jolly, slim=unhappy" or vice versa. It depends on culture.

The report that this newspaper article was based upon is:

• Patrik K. E. Magnusson, Finn Rasmussen, Debbie A. Lawlor, Per Tynelius and David Gunnell, Association of Body Mass Index with Suicide Mortality: A Prospective Cohort Study of More than One Million Men, Am. J. Epidemiol. (1 January 2006) 163 (1): 1-8. doi: 10.1093/aje/kwj002

They found that there was a

strong inverse association was found between BMI and suicide.

and that

weight loss as a consequence of mental illness does not explain the BMI-suicide association

Now, suicide is not the opposite of happy, so the newspaper report was a bit off-the-point.

One of the authors was involved in a later study:

• Ottar Bjerkeset, Pål Romundstad, Jonathan Evans, David Gunnell, Association of adult body mass index and height with anxiety, depression, and suicide in the general population: the HUNT study. Department of Research and Development, Levanger Hospital, Health Trust Mid-Norway, Levanger, Norway. American journal of epidemiology. 01/2008; 167(2):193-202. DOI:10.1093/aje/kwm280

Raised body mass index is associated with an increased risk of depression but reduced risk of suicide in men and women.

That's an unexpected result - obese people are more likely to be depressed, but less likely to commit suicide. The authors acknowledge this needs more clarification.

Remembering that depression ≠ unhappy and suicide ≠ unhappy, and now we see that depressed ≠ suicide.

So, neither of these directly address the question, despite being the basis for it.

• Graham, Carol and Felton, Andrew, Variance in Obesity Across Cohorts and Countries: A Norms-Based Explanation Using Happiness Surveys (September 2005). CSED Working Paper No. 42. doi:10.2139/ssrn.1024823

Well-being is a better match to happiness than (non-)depression or (non-)suicide.

This was an interesting study that showed that there were, perhaps unsurprisingly, cultural norms and socioeconomic factors involved in the question. Obesity is correlated with being poor in the U.S., but (apparently) not as much in Russia.

In the USA:

Poor whites have higher obesity-related well being costs than blacks or Hispanics. Respondents in the top income quintile who are obese and those who depart from the weight norm for their profession also suffer higher well being costs than the average. Stigma seems to be higher for those in higher status professions. We find modest evidence that causality runs from overweight to depression rather than the other way around.

In Russia:

obesity and well being are positively correlated. The relationship seems to be driven by the prosperity that is associated with obesity rather than by the excess weight per se, and we find no evidence of stigma.

• Marina Selini Katsaiti Obesity and happiness Applied Economics Volume 44, Issue 31, 2012, DOI:10.1080/00036846.2011.587779

Looking at some additional Western cultures, This study looked at well-being, which is more closely related to happiness.

Results indicate that in all three countries [Germany, UK and Australia] obesity has a negative effect on the subjective well-being of individuals.

In summary: the answer didn't come from the original paper, and is far more complicated than the newspaper might suggest with its cherry-picked anecdotes.

The relationship between poor well-being and obesity is culturally dependent and socioeconomically dependent; the link (and stigma) is stronger in Western countries than Russia. There is a relationship between depression and obesity in the USA, and seems to be more likely to run from obesity to depression. There is an inverse association between obesity and suicide, despite the depression figures the other way.