Evidence?
Actually, the research I cited indicated that the women in women's shelters don't perceive trans women in this way - and that's not just the testimony from the women who run women's shelters, but also attested to the fact that they are currently legally allowed to exclude trans women on the basis that they are trans if they do make the other women uncomfortable and not a one of them has ever had cause to use that exemption.
So, yes, all the actual evidence that I've seen suggests that most women are fine with the idea of trans women in women's shelters, and that is borne out by the actual experiences of people running women's shelters. Which, in case I need to remind you, already operate on a basis of self-identification.
"Well, my mate thinks..." and "it stands to reason..." are not counter-arguments, I'm afraid.
If the general case isn't working, switch to the specific. If the specific case isn't working, switch to the general. Above all, keep switching so that there's never an actual claim to be defended.
I bring up the point that allowing sperm producing people into women's only spaces erodes the system of women's only spaces, and you ask for evidence. Huh? It's kind of a definition thing. Oh, wait. There's the problem. You're not much into definitions. Definitions of words like "man" or "woman" just make the issue so cloudy.
People don't perceive transwomen "in this way". What way? Bigger, stronger, with penises? If they aren't perceived that way, then soemone needs better perception, because transwomen tend to be bigger and stronger than "cis-women", and they have far more penises. Or do you mean that the transwomen aren't actually perceived as creating discomfort? Well then, why are we having this conversation?
But of course, transwomen in female only spaces do cause the women in those spaces discomfort. Show me a transgirl in a girls' locker room, and I'll show you a set of girls trying to keep her out. Some will say "him", and some will say "her", but they will all agree that the transgirl has a penis, and the girls would prefer it if they could use their locker room as it was intended without the presence of the sperm-producer. It's almost as if the girls think of the penis as not just a body part, but as a very significant body part. It's almost as if genitals matter to people.
So the people who run women's shelters report that they could accept trans-women, but they don't, therefore....what conclusion do we reach? That the women in the shelter don't have any issues? Then where comes the controversy? Does this apply to prisons and sporting events and bathrooms too? Because it seems to me that an awful lot of women are arguing about this sort of thing. Somebody must have some sort of opinion on the subject. I think an awful lot of women have issues with males in their female only spaces, and that's why it's a controversial issue.
Quick. Say, "Evidence?" It makes you look smart.
ETA: As for shelters, specifically, I'm all for giving the people who run the shelters discretion in the matter. For prisons, what's most important is that males cannot be in a position where they can overpower and rape females. Also, women should only have to undress in the presence of persons of the same sex. That's "sex". The same applies to men, but it's not as important.