Just as the rights stance is the birth certificate argument. In reality there will be a compromise on both sides and definitions for "gender identity" and "trangender" will need to be defined. Since nobody is going to be monitoring bathrooms it really is pretty much a political and pointless debate.
My prediction is there will be a point in a persons transition, after consulting therapists and doctors, after living as the opposite sex in dress and appearance, that they will legally be able to change their gender on their ID. I don't think this will necessarily require surgery. These individuals will then be able to use the bathroom of the gender defined on their IDs. That's my guess and an approach that I think is fair to both sides.
You think MDs are going to sign off to put men with penises and testicles into women's shower facilities, dorms and bathrooms eh? I think the MD's would fear lawsuits and shy away from doing that.
There's no compromise, because there's no medical evidence supporting "transgender" as a real thing. All "transgender" really means is "the medical community is afraid to confront these particular delusions in the mentally ill, so they're calling it something normal and healthy so they don't have to face the wrath of the Church of LGBT for refusing to play along". That's what "transgender" means. And that's not a viable explanation enough to force women to deal with dudes in their showers and locker rooms.
Remember, we have the 17 million rape survivors' needs to consider. And be considered they will. So, good luck with your "compromise"! And BTW, women are always aware of who they're with in the bathroom. The situation will be monitored. Even if it means parking their husband outside keeping watch on the door to apprehend any dude in a dress trying to walk in. There will be monitoring. Vigilante style.