Do you actually have any evidence that men are very likely to or do attack women when they're in the same public bathroom? What are the figures, the research, the study on that?
Now, I know that trans women sometimes get messed with in the men's room (I had a trans woman friend who was verbally hassled for looking feminine even before she started estrogen therapy), but they also get messed with in women's rooms. For example:
Beating of Chrissy Lee Polis - Wikipedia, the free encyclopedia She was beaten into a seizure for trying to use the women's room, by
women. Not by men. I find that it's usually women who are most likely to react with panic or violence when a person viewed as male or masculine comes into a women's room/space. I've heard this, not only about trans women, but butch cis women also get harassment in the ladies room.