I am not sure about requiring trans women be allowed in female bathrooms.
Who decides who qualifies as being sufficiently "trans-woman"? Must they have engaged in reconstructive surgery? Be taking female hormone therapy? Must they dress according to "average" standards of female dress?
What if the person wants to be a transwoman, who dresses like a man? The possibilities are endless and crazy.
This is why some places have simply decided to have a third bathroom option, that is gender-neutral.
The third space is an ideal solution. Nevertheless women find to their cost that even when such spaces are available,
men LARPing their pornified fantasy of femininity insist on colonising the women's bathrooms regardless. This is because their primary desire is for "validation" of their belief that they are women. If you don't know about autogynaephilia, you're not up to speed with this debate.
Human beings in general are extremely good at telling which sex other human beings are, and women are better at it than men. I do not need to see someone's genitals to know if they are male or female (we allow for occasional sincere mistakes which can be solved with goodwill on both sides).
I know when someone is male. I do not know whether or not he is taking hormones or whether he has had genital surgery. I don't care what he's wearing. In many cases, a man LARPing womanhood is more alarming than any normal man. (I don't have a problem with an ordinary man who has wandered into the wrong bathroom by mistake, I know he isn't weird or a thread. I do have a problem with the weirdos in lipstick in that film I linked to.)
What I (and most women who have thought this through) is giving any male the
legal right to be in a female sex-segregated space. What are we supposed to do to check that a man we see in there has that legal right? Ask him to strip off to prove he's had his penis and testicles amputated? I don't think so. Ask him to show his hormone prescription? I don't think so. Critique his wardrobe? Really?
The de facto result of giving
any male the legal right to be in our intimate spaces is that we can't keep any male out. When challenged he will merely claim to be one of the select group with that legal right and accuse us of transphobia and hate speech. We lose the ability, which we value greatly, to police our intimate spaces for the presence of males and invite any male who wanders in there to leave.
As I said only a few pages ago, if someone presenting like Blaire White comes in, I almost certainly won't notice that he's male (even though he is a fully intact man - the illusion is very good). So no problem. If someone presenting like Debbie Hayton comes in, I will notice, and be a little creeped out, but if he simply goes into a cubicle, comes out, washes his hands and leaves without attempting eye contact or conversation I will tolerate his presence. However, if there is another woman there with a lower tolerance threshhold for men in women's spaces and she asks him to leave, I would expect him to do so immediately without escalating this into a confrontation.
That is how things worked until very recently. There was a degree of tolerance, but men had no legal right to be there and a woman could ask a man to leave and she would be in the right.
You want to give men - in effect
all men who are prepared if necessary to claim to be transwomen - the legal right to colonise our spaces and remove our ability to police them so as to keep predators out while our daughters are dealing with their first period.