why is every single institution of the Patriarchy currently at war with trans people
Not that I've seen. The majority seem at present to bend over backwards to placate trans people and attend to their grievances, and demand (often with menaces of one sort and another) that all women must at all times do the same.
The real question is, of what do you think this "war" consists? Actual harm -- lower wages, for instance, and reports of crime against them being ignored? Or failing to pander to someone's every stray feeling?
But as we have seen, the chance of a woman being assaulted by a trans woman, or a man pretending to be a trans women, in women's spaces is millions to one.
I haven't actually seen that. Can you give a link to the data, please?
The rate of violent crime, including rape, committed by trans women (where is still recorded as being by trans women and not women, which is in itself a scandal but not relevant here) is the same as the rate for similar crime committed by heterosexual normative men. There is no reason to think that this will be altered by the realisation "oh, no, I mustn't attack this woman I have got on her own -- we're in a women-only public lavatory, it would be wrong", is there.
Men's hostels and refuges won't let trans women in because they know they wouldn't be safe.
This is a problem of male violence, not female violence. Take it up on men's discussion boards, where men talk with men; it really isn't needed here, where in that respect you are definitely preaching to the choir.