Well, according to the website, the gravitational pull due to the Shapiro Effect decreases with the inverse of the logarithm of distance, not the inverse of the distance. And this has supposedly already been verified by observations of the Mariner spacecrafts. If true, it would mean that gravity would still have an effect even at much greater distances.
I don't quite see your point about the correlation of time for particles with mass and the light. The gravity is affecting the light after it has left the emitter and continues to affect it as it travels through empty space, just as the expansion of the BBT is affecting the light as it travels through space. Neither of these effects on the redshift have anything to do with the light as it being emitted from the source. So I fail to see how you could distinguish which of these two possibilities caused the redshift based on particles being emitted by the supernova. The gravity strength of the source may affect the redshift slightly as the light is being emitted, but I would think this effect would be minimal and is not what the Shapiro Effect refers to. Any further information on this correlation or a link to a site falsifying the Shapiro Effect would be appreciated.
One more thing to consider, is that if the Shapiro Effect is something to be reckoned with, and if the BBT is also considered correct, then the Shapiro Effect could have a drastic effect on how we interpret certain observations. For instance, if we use the current estimates of the expansion rate as being correct, it would mean the universe is much younger than what most cosmologists say it is. Furthermore, any observations of events early in the formation of the universe would be even more profoundly influenced by gravity due to the universe being much more compact at that time and thus a much larger Shapiro Effect. So in this case, the whole model of the universe would likely fall apart because there wouldn't be enough time for the universe to develop as we see it. But, on the other hand, with the Shapiro Effect it would mean that the expansion rate could be much less to achieve the redshifts that we observe, and this would give us a much older universe.