So do you support letting trans-"women" play on women's sports teams and use women's bathrooms?

I think sports teams and sporting organizations should make their own decisions. I don't know what's actually best overall; I think transgender people are rare enough that it wouldn't make a big difference in practice to most . My guess is that the best policy for smaller informal sports teams and organizations is to let 'em in, that the best policy at the highest levels where a lot is at stake is to say women's teams/competitions are only for people who are anatomically female by some criterion or other, and in between I'm less sure but lean towards a let-'em-in policy in the absence of compelling evidence that it would do actual harm.

I think that if someone is generally presenting as female we should let them use women's bathrooms if they want to. The obvious objections to this seem to be (1) ewww (which I suggest is not an argument) and (2) that this introduces a danger to women from predatory men dressing up as women in order to sneak into their bathrooms. I find #2 unconvincing because when I try to imagine scenarios where there's an actual difference in the harm done I can't think of one that's actually plausible, and because whatever bathroom policy we adopt there are going to be trans people and they are going to need to use bathrooms, and there are obvious risks of harm from trans women trying to use men's bathrooms too.

Of course, for purposes of those Bayesian deductions trans-people are much closer to their biological than their claimed gender.

I don't think that's clear at all. As I've said before in this discussion, what counts as "closer" depends greatly on context, and for many purposes someone who looks more or less female, presents as female, and considers themself female is "closer" to stereotypical-female than to stereotypical-male whatever is in their chromosomes or their pants.

That he's in the similarity cluster labelled "women"

Someone in this situation is some way from the centre of either the "women" or the "men" cluster, regardless. It seems to me that in fact there is no such thing as the similarity cluster labelled "women" because (have I mentioned this already?) there are any number of similarity clusters corresponding to different notions of similarity, and different notions of similarity are called for in different contexts.

If you pick some particular notion of similarity based on (say) gross anatomy, sex chromosomes, hormone levels, and ability to beget and/or bear children, then indeed our hypothetical person is in the "men" rather than the "women" cluster. But do you really think there's a delusion there? If you ask, say, Bruce->Caitlyn Jenner "What chromosomes do you have?", the answer might be "It's none of your business" or "Who cares?" but it won't be "XX, of course, because I'm a woman".

It seems to me that the actual difference between you and, say, Jenner is a disagreement about what notion of similarity to use. How is that a delusion on a par with thinking you're Jesus?