I don't think you could send a TW to a men's refuge, though?! Maybe a mixed sex "domestic abuse" refuge or similar, but I understand that in itself would have problems.
I imagine the main issue is preventing abusive behaviour within the refuge itself. I'm sure very few people would have an issue with a "genuine" TW in women's space, but the issue comes with the (presumably few) men who lie, basically, to gain access or advantage. But the whole self ID thing I think removes any safeguarding around that... in that there is no recourse to saying "no, you don't qualify to access our service". If everyone had to produce a birth cert or something on entry, and a formal GRC if the sex on that is different to the service sought it would at least be some kind of proactive thing to avoid the risk of "mistaken" admission.
Obviously someone (of either sex and whatever presentation they choose) abusing another resident can be thrown out - but presumably the issue is before that. If a complaint is made about someone, that residents feel uncomfortable or whatever, nothing can be done about that until actual abuse occurs. Which seems rather dodgy to me.