But ultimately, I believe that the right of trans women to have a space to shower / pee / get changed is greater than the right of non-trans women to be protected from the occasional sight of a penis.
But if all they're doing is having a shower and getting dressed, then what's the damn problem?
So your position is that we don't need single-sex spaces at all. Fair enough, it's good to understand that this is your perspective.
There are lots of reasons why women-only facilities were created (after not previously existing) but there are some people who don't believe it's necessary. It sounds like you're one of them.
Although I've never experienced sexual assault in a changing room or toilet myself, I support single-sex facilities based on the statistical evidence that they reduce (though obviously don't entirely remove) this risk.
Unfortunately the risk will never be zero. There will always be males who are so determined to break boundaries and gain access to women in order to satisfy sexual urges. But if everyone knows that there are single-sex facilities and no males should be in the female ones (unless it's a cleaner, with a sign clearly displayed so that women can choose to enter or not), that goes a long way to reducing risk. If the boundaries are blurred for any reason, and some males are allowed in, that risk (by statistical definition alone) goes up.
And for anyone who believes that TW wouldn't do that kind of thing, here's a list of 236 that did:
https://grahamlinehan.substack.com/p/this-never-happens
The list isn't specifically about changing rooms and toilets, it's a list of males who identify as women who have predated on women or children. Given TW are such a small proportion of the population, the odds that some will do this kind of thing seem disproportionately high from these successful court cases alone. There will presumably be others who weren't caught.
The main problem here is that, until a sexual assault is committed, there is no way to tell which TW is simply trying to shower/pee/get changed and which is working up to something like this. The most obvious answer is to have single-sex spaces without any blurred boundaries. If TW experience discrimination when using male facilities, they can campaign for access to additional third spaces. In some cases those spaces may already exist (e.g. accessible toilets that welcome disabled people, parents changing babies' nappies etc) but they would need to prove that they weren't putting too much strain on those existing facilities. If they were, they could campaign for more provision. Equally, they could campaign for males to be more inclusive in men's facilities.
Edited for typos.