Actually that statement was meant to convey both the bigotry of the area and the likely puritanical approach to nudity. Let's be serious though, just look at many of the responses in this thread. I am quite confident that bigotry against transsexuals plays a strong part.
I'm not asserting it's the ONLY factor, but to think that this is happening absent anti trans bigotry strains credulity.
Where did you get the idea that she is comfortable with using male facilities? From my experiences with trans friends and acquaintances I strongly suspect this is not accurate.
I simply don't think that one shitty aspect of our culture (body shame) should be used as an excuse for another shitty aspect of our culture. My guess is that this individual is far more afraid of what other people think of her than they are of the reverse.
The bigot card is really old. No one is protesting her being at the school. No one is dragging her behind their truck. They just don't think someone that is technically a boy in a dress and wig should be changing in a open room with a bunch of teenage girls.
I think it would be quite a bit different if she was taking hormones, grown out her hair, etc. But even then not many dads are going to want someone with a dick changing in front of their daughters and I doubt many girls want it either. Even if they were fine with her seeing them naked, the dad's aren't going to want their daughters seeing a dick every day.
Further, if all it takes to get in the girls' locker room is a dress and wig, there will be other guys doing just for the view. As an example, a company my wife used to work for allowed shorts, because some guys complained that women could wear dresses and they couldn't wear shorts. Their boss said "well, you can wear a dress too." So a large group of them did for months until they changed the dress code. It isn't that hard to wear a dress for a decent payoff.
Finally, if you are truly above body shame, post naked, timestamped pictures with identification of yourself and girlfriend/wife/sister. I'll be waiting over at /B/. If you are uncomfortable doing that, why should a bunch of high schoolers basically have to do the same thing every day in front of someone that makes them uncomfortable?
I am not sure what the true final solution will be for these matters, but I think offering gender neutral facilities is an effective solution for the time being. Maybe someday we will all be beyond modesty, and we can have 6 year-old girls showering with unrelated 60 year-old men, but in the mean time I think we need to come up with a practical solution that is as fair as possible to everyone involved. It sounds like this school has tried to be fair by offering a gender neutral facility.
Edit: BTW: I fully support transgender, I have zero issue with them, and I hope making the change either emotionally and/or physically will bring peace to them. But I also don't think modesty and prudeness are evil ideals and respect people who value those ideals.