So what happens if we require trans women to use male facilities in public places? Do you think they will be safe in a designated men's bathroom or shower if they are publicly presenting as female, penis or not (because attempts to legislate on this issue use sex at birth, regardless of the stage of transition)? The effect of bathroom regulation is to create a situation that excludes trans people from all public facilities, and by extension the ability to exist in public spaces, because they can't venture out of the home for longer than they can control their bladders.
It's maybe worth discussing the background to this in the UK, which I think is rather different from that in the US.
Here everyone -- in trans matters as in everything else -- enjoys the protections of the Human Rights Act and the European Convention on Human Rights, regardless of whether a Gender Reassignment Certificate (GRC) has been issued.
If a certificate has been issued, then that's an end to it -- the GRC has the same legal force as the birth certificate which it supersedes -- but even when there's no GRC, trans people still have a right to privacy about personal and medical matters -- so no outing them by misgendering on Twitter or anywhere else -- and still enjoy the same protections against harassment and public humiliation as does everyone else.
At the moment,
reforms to the now very outdated Gender Recognition Act 2004 are under consideration, which a view to simplifying the system, making it a lot quicker, easier to use, and less medical, and relying more on self-certification.
However, and I think this is the point that tends to be missed, no revision of the GRA can supercede the HRA and the ECHR, so any new regime must, as a minimum, continue to afford both trans and cis women the same protections they enjoy at present, even though the protections will be somewhat more difficult to enforce for someone who does not yet have a GRC.
So I think that, however much some people may wish to import "bathroom wars" to the UK, it's not going to happen -- regardless of any changes that may be made to the GRA, the people responsible for public bathrooms (and for changing rooms, prisons and so on) will still have to ensure that the qualified rights of all the people using their facilities, cis or trans, are protected, with the supervision and support of the courts if necessary.