> He has his entire reputation staked on him being right about his own theory
You may be thinking of someone else. I'm not aware of Carroll having "his own theory" that he's "staked his entire reputation on."
In any case, he's explained his position, even in the quote I provided. He's simply describing what general relativity says about this. If you have some issue with it, you can respond on technical grounds, without needing to resort to ad hominem.
> He likely believes in the Big Bang too which is an absurd, and disproven theory.
Big Bang is not "disproven" - the Lambda-CDM model of the Big Bang is still known as the "Standard Model of Cosmology," and nothing has replaced it as a consensus among mainstream scientists.
> There are far too many mature galaxies that are far too old for the Big Bang theory to be correct.
The issue of galaxy ages has a number of reasonable explanations in the context of Lambda-CDM. One of them could simply have to do with age estimates, which is an area that suffers from limited observational data that's difficult to interpret. Recent work on the Hubble tension may help resolve that, e.g.: https://news.uchicago.edu/story/new-webb-telescope-data-sugg...
The idea that galaxy ages "disproves" Big Bang theory is at best, a misunderstanding of pop science hype, and at worst pseudoscience. That's not how science works, especially in cosmology where most evidence is very indirect and subject to interpretation.