You judge too quickly.   The only transgendered person I've know personally, is a man who was born with sexual equipment of both genders.  His parents picked to raise him female, he chose to be male at great expense and deadly risk of transformative surgery (30 years ago).   He used the men's restroom in our very blue collar factory, without conflict during the time that I knew him.

What determines 'maleness' for you?   Size of reproductive organs, hair growth, muscularity, height, weight?     Are some men more or less male based on their physical characteristics?    Can anybody be judged by physical appearance alone?

The conflict is not as clear cut as you, or the administration, is making it.   It would seem that a bathroom can be designed for more privacy, and I suspect that's mostly true for women's restrooms already.