Replying to all who have replied to me: I read the mainstream press, and I haven't read anything about these pervs doing such things. Surely if you go looking for online spaces where pervs hang out, you will find pervs?!
I agree that transwomen should be more sympathetic to women. If I were a transwoman, I would not want to upset anybody, and I would use single-space loos and changing rooms wherever possible. I would use the disabled loo rather than the women's loo, and I'd change there too, if we're talking about a swimming pool. Yes, I know they're meant for disabled people, but if there's no disabled person in sight, then I don't think it matters if it's used by a transperson for a minute or two. I also would order undies online instead of making women uncomfortable in lingerie departments. I would go out of my way to avoid pushing into women's spaces, while also trying to make sure that I didn't get beaten up in a men's space. i.e. I'd probably feel more comfortable as a transwoman using the male loos at the theatre rather than the football ground.
There might well be transwomen who do all this, but you wouldn't know, because by definition, they're more invisible.
I dunno, I just tend to think positive and believe the best of most people. And I believe that most transpeople are genuine and don't want to upset women. I think it's quite possible that posters here are focusing too much on bad actors, who aren't the majority.
ETA: About the breastfeeding, I agree that there are too many unknowns about the milk that's produced. But since transwomen produce so little anyway, it sounds as if it'll be a non-issue. I'm sure that the majority of transpeople who are parents use formula, like gay male parents, for the very reason that we don't know the longterm effects. Again, we're focusing on an outlier.