The quote in the OP from the creation article is an accurate summary of a mainstream Science Daily article: Big Bang's Afterglow Fails Intergalactic 'Shadow' Test:
If the standard Big Bang theory of the universe is accurate and the background microwave radiation came to Earth from the furthest edges of the universe, then massive X-ray emitting clusters of galaxies nearest our own Milky Way galaxy should all cast shadows on the microwave background.
In turn, the Science Daily article is based mainly upon:
The Sunyaev-Zel'dovich effect in a sample of 31 clusters: A comparison between the X-ray predicted and WMAP observed decrement, Astrophysical Journal, Vol. 648, p. 176-199. (alternative link to pre-prints)
One vital test of the present cosmological paradigm is the
search for scattering of the CMB by foreground structures such
as clusters of galaxies. Such observations can provide important
information both about clusters of galaxies as well as basic
cosmological parameters like Ho. For the CMB, scattering arises
from the Compton interaction with free electrons in the hot
(X-ray temperature) plasma of clusters of galaxies, which removes
Rayleigh-Jeans blackbody flux in the direction of a cluster,
and leads to an apparent decrease in the CMB temperature,
a phenomenon known as the Sunyaev-Zel’dovich effect (SZE).
By now, the degree of SZE is highly predictable for many clusters
of galaxies, because their hot intracluster medium (ICM)
properties are well-measured by X-ray satellite missions.
In summary, it is through the first detailed radial profile comparison
between X-ray and microwave observations that an apparent
sample-wide discrepancy between the expected and measured
levels of SZE from some of the best known clusters of galaxies
See also Detailed X-ray/WMAP comparison for a sample of 31 nearby galaxy clusters - incomplete Sunyaev-Zel'dovich silhouette and the question of the CMB distance scale the abstract of which says:
A resolution of this discrepancy between predicted and observed decrements have potentially extreme ramifications for our interpretation of the CMB. One is forced to conclude that either the CMB is non-cosmological, or there are issues with the WMAP data itself which must be taken into account when interpreting the CMB emission.
See also the Phys.org article Big Bang's Afterglow Fails an Intergalactic Shadow Test:
The apparent absence of shadows where shadows were expected to be is raising new questions about the faint glow of microwave radiation once hailed as proof that the universe was created by a "Big Bang."
which offers another popular science summary.
Both the creation article and the Science Daily article quote Dr. Lieu as saying:
Either it (the microwave background) isn’t coming from behind the clusters, which means the Big Bang is blown away, or … there is something else going on
The title in the OP isn't representative of this "either ... or" statement.
In other words, there are alternative explanations, no one quoted in the OP is saying that the research "disproves" the big bang theory.
Dr. Lieu's publication was from 2006. Looking at articles citing to Dr. Lieu's is the best way to see the current status of this research. The big bang theory is not considered disproven.