I'm not saying other people shouldn't have a voice on this topic, but that discussion is a topic that is based in transphobia. If a woman feels uncomfortable sharing the bathroom with a trans woman, it's because she has a predisposition to fear and feel unsafe around trans women.

Contrast it with the past, where men used to feel uncomfortable with women in typically male dominated spaces. I remember seeing an old news video where a female reporter was sitting in a bar typically frequented by men. She asked them how that made them feel and all the men she asked made a point to mention they felt uncomfortable with her in the bar with them and they would rather her leave than share the bar with her.

This is the same argument you're presenting. It would seem ludicrous now as we've moved past that point (for the most part, it's not a 100% solved issue sadly), but at the time it was seen as a legitimate argument, as sexism was still rampant at the time and it was the norm for women to be excluded from male spaces.

There is nuance to your presented argument in that the space that's in question is a restroom, which is a more private and intimate space, but the nature of the argument is still based in transphobia and I think should be presented as such. It's the root of the issue and the aspect that needs to be addressed before any headway can be made on the topic