The onus should be on trans people and their allies to campaign for this provision, rather than moving in on the spaces that women have managed to secure for themselves
I'll come on here boringly to say, once again (once more with feeling), that access to safe, private, and secure, public lavatories was an important campaign in the 19th century to enable women to enter into public space.
I'm not sure just how much women today realise how constrained "respectable" women were in the 19th century. The expectation was that a "respectable" woman would not leave her house on her own. And if she did, she was not respectable, with all the implications of that. For example, the Contagious Diseases Act enabled police in port towns (eg Plymouth, Southampton) to forcibly incarcerate women they suspected of being prostitutes and forcibly inspect them gynaecolgically for signs of STDs (look up the instruments they used. The point here is that almost any woman walking alone could be picked up by authorities in this way. So working-class women, walking to & fro their jobs were particularly at risk.
Because the "logic" was that any woman in public was - by reason of being in public without a man - therefore not "respectable."
So, that's an example of the legal punishments of women in public space.
I am old enough to remember that it was a regular puny excuse for not employing women in so-called "masculine" jobs (eg engineers, architects, doctors, university lecturers, manual labourers, carpenters - anything that paid well, really!) that there were not enough toilets. And no room to build new ones.
this is living memory and lived experience.
So don't dismiss women who object to the undiscussed slide into mixed-sex lavatories. Safe, private, secure facilities for women are essential.
Not having them has been used for the last couple of hundred years as a big reason why women should not participate in public life as workers, employees, and in leisure activities.