Aw man, I'm so late to this discussion!
Personally I think that there's no reason why either group's feeling on whether a transwoman is a woman or not needs to win out over the other.
I mean I have my opinion on that, it's credible enough for me and others who hold this opinion, but those who think otherwise have equally credible reasons.
We could argue it in an ideological sense until the cows come home!
But I don't think, in practical terms, that needs to matter.
Because here's the thing; we don't exclude anyone from anything for no reason at all; that's discrimination. But what we do do is lawfully and rightfully exclude people when there is good reason. If there's a good reason to exclude males from something so that it is "female only" that sure, crack on. And similarly if there is good reason to exclude someone who has the protected characteristic of "gender reassigment" from something then we can do that too!
If for some reason there is situation where it would not be appropriate for transwomen to be socially and legally treated as women (i.e. in a woman's shelter for example) then we don't need to say "they're men!" in order to do that; we can simply say their gender reassignment poses a problem, and exclude them as required.
Ultimately I can't see that it matters much whose ideology is more credible. Society and the law doesn't seem to rely on it, from what I can tell.