You are brave to post this @Rilkescat , but I agree - mostly. Whilst I absolutely support womens safe spaces being kept for female born women only, I must admit I have NEVER experienced a 'man dressed as a woman' trying to come into a safe space for females... be that toilets, changing rooms (swimming pool and clothes changing rooms) and womens support groups/breastfeeding groups etc.
I am not saying it never happens, but I have never known it happen, and I don't think anyone I know has ever seen it happen either. If they have, they have never mentioned it anyway! I have to say I don't know how I would feel about it. Depends on the situation, and what sort of group it was. If it was a womans 'support' group, then I don't think I'd be happy with it.
Not exactly the same but in a similar vein... A woman in the area I live, used to have a 'Womans' group at her house, and invited a few friends, a few women from the Church, and a few neighbours - including me. We chatted about all sorts of issues from childhood trauma, to adult trauma, and depression, and loss, and periods, and menopause, and children and parents etc, and we had lots of laughs and wine too. (About 15 of us between 40 and 70.)
One woman (who invited herself when she heard one woman from the group reminding me about it this Wednesday,) asked - the second time she went - if her HUSBAND can come next time, as 'he wants to come...' Different circumstances of course as he is not transgender, but it was a WOMANS group. The leader said 'no, it's for women, specifically for women.' The woman in question grumbled and bitched to me and said 'she is breaking the LAW excluding him!'
I said 'she is really not as it's her house. Her house, her friends and neighbours and her group. Not a public group.'
As I say, different circumstances but similar, because it was a man trying to shove his face into a womans group. The woman came the next week, but didn't come the following week. The lead woman told everyone what she had said. Not one single woman wanted him there. Why this bloody stupid woman wanted her fucking husband there remains a mystery. Moreover, why did he want to COME? 
The woman came back a week later and asked AGAIN, and said her husband regarded it as unfair and probably against the law to not invite him. The leader said he is quite welcome to start his own group, and the woman mithered and whinged so much, that 2 hours into the evening, the leader told her she was not welcome anymore, and she needs to leave!
Women have been excluded from enough MENS stuff over the decades/ centuries, so these entitled men can fuck off. We are entitled to our own groups and safe spaces. And that includes personal private intimate shit like womens support groups and changing rooms and toilets.