Which is why I don't think you should have to prove it. Just use the bathroom you're comfortable with.
The problem here is that this flies in the face of the entire reason bathrooms were segregated in the first place - because, by and large, arbitrary cultural programming though it may be, a large number of people are not comfortable going to the bathroom/changing in the locker room in the presence of the opposite gender. And since there is no way to reasonably require "proof" of transgenderism, it means that
this happens, in accordance with and protected by law.
But the intended effect is to force people to use the bathroom they don't want to, aren't comfortable with, and are likely to take abuse for using. Its solving a problem that barely exists and replacing it with one thats depressingly common.
As I said, if the first law hadn't overstepped the government's role in the first place, we wouldn't be in this mess. THAT law attempted to address an uncommon problem (by your own words, everybody was "just fine" before all the law shenanigans) by creating a new, easily more potentially numerous abuses. Remember, transgendered people number less than a tenth of a percent of the population.
Also, from a pragmatic standpoint, I somehow doubt this does anything to positively impact the so-often-used as a fallacious argumentum ad passiones example of bathroom violence against the transgendered. It's a well-intentioned blunder that's just made everything worse, and in the long run, probably won't be particularly effective. It'd have been much simpler and more effective to simply make all the public restrooms unisex, and really get started deprogramming the culture from thinking we need to hide from each other to pee. Some places in Europe don't have this problem because they don't stigmatize coed nudity.
So, in my opinion, the easiest solution is, instead of having a men's room and a women's room, have a regular (unisex) restroom and a "family" restroom for people with kids, with facilities appropriate for parents with small children (larger stalls, shorter toilets, changing tables, etc). That way, if you're an adult, you still can get the more faster, efficient room, and people who are paranoid about strangers oogling their kids still have their own place to go.
And you don't have to stomp anyone's toes by "letting men in the women's room" by way of refusing to have a physical definition of what constitutes either one - because this way it no longer is important in a practical sense. It also requires a lot less renovation than converting buildings built around a 2 bathroom paradigm to 3 bathrooms.