I may occasionally use sarcasm enough to the point that I might better mark the posts that aren't sarcastic to save time. So consider this a /s as a heads up...
Unless you have been in this position, you really don't know.
Transgenderism is a very complicated issue. It isn't "yes" or "no". It's a sliding scale. Some might consider themselves transgendered, but might in fact only be suffering from something else that may only be a temporary mental condition that can be remedied with counseling. Others may in fact have identified as a gender different from what their anatomy may suggest their entire lives and should transition. And others still could be classified as "inter-sexed", or having organs from both genders. Ideally, everyone would all fit neatly into the X or Y column with no variation. But it doesn't always work that way. The existence of inter-sexed patients alone proves that "god" can and does make mistakes and we as a society need to deal with the consequences.
The media hasn't been very helpful in this regard. Often shooting for the most polarizing and sensational angle possible, usually seeking out the extremes, passing them off as the rule and stringing things together in the most damning and insinuation filled faux accusations imaginable. The wide majority of those that are transgendered just want to live in a world where they can be accepted for who they are in the inside. Yet the takeaway from one of the stories linked is all "The nutzers trannies are pushing us around and if we don't push back, Big burly hairy men are going to get naked and rape your 6 year old daughter".
This kind of journalism isn't completely unlike what Putin has pulled in telling gays coming to the Olympics "You can be gay, but stay away from our kids", as if to use innuendo and insinuation to tie being gay to being a pedophile. That's too bad. And media that sticks to facts and allows a viewer to make their own determination has gone the way of entertainment and the advancing of agendas under the false guise of journalism. But we have to expect better if we consider ourselves to be mature adults that live in a society that can deal with these kinds of matters seriously, without caving in so readily to the outrageous.
In any event, the fairest option I can think of is to provide people who identify as being transgendered with a road to using their preferred bathroom, but make it a little difficult. Have the school require letters from a psychiatrist, general practice and endocrinologist certifying that this patient has shown strong signs of being transgendered and are taking it very seriously. That takes time and effort and serves as a deterrent to those men who think they can just come out and say "Ok I feel like a girl now, let me in their locker room" and hop right in. Consideration has to be afforded to both points of view to some extent or you risk undermining progress.