I think where the changing room issue is going to end up (maybe a long time from now, after all the extremists have worn one another out) is that trans women will be permitted to use women's changing facilities, but not permitted to show a penis.
If that restriction makes it a bit more difficult for the trans woman to use the facilities, forcing her to use a stall or don a covering skirt while changing undies or wear an extra garment in the shower, then that's regrettable, but all kinds of things are more or less difficult for all kinds of people. Short people have more trouble reaching items on high store shelves, sometimes forcing them to use a step ladder or a reaching tool or ask for assistance. We don't consider that a gross injustice or a failure of the system (unless the needed assistance is unavailable).
It's important to note that no one in recent times (up until trans rights became a public issue and the rules became haphazard) has had the right to show a penis in women's rooms. It's not equivalent to the "gay men have the same right to marry a woman as straight men have" argument, because no one in our society, male or female, has ever had the right in question. If you think women might somehow have had the "assumed" right to show a penis in the changing room, consider the likely reaction, throughout the past century, to a cis woman wearing a realistic prosthetic penis visibly in a women's changing room. Or a visible strap-on dildo. Or just taking out a photograph of a penis and saying, "hey, look at this!" Or quietly taping such a photograph to the wall. Would the other women present, or the authorities, answer any objections with, "she's a woman so she has the right to do that"? Of course not. If the woman doing so were to persist after objections, such actions would be addressed as indecency, or disturbing the peace, or creating a public nuisance. And that's not even for a real present functioning penis.
I don't expect either of the main sides in this argument to like this solution. Some will call it transphobic (how dare I expect trans women to meet an extra requirement in a women's room), others misogynistic (how dare I expect females to tolerate male bodies in the changing room, however discreetly presented), and still others just too conventional (what, I'm holding onto the archaic notion of gender separated facilities?). A good compromise leaves nobody happy, but everyone able to muddle through.
Questions concerning women's sports, prisons, shelters, etc. will need their own solutions, of course.