I haven't heard of this "Prove yourself" brigand you speak of, myself.
I used to have a Facebook, maybe six years old, that wasn't in my current legal name and I was never stopped or questioned, emailed, nothing. Eventually it got deleted from inactivity, but nonetheless if you set it to your preferred name straight off, I don't think that they have any reason not to believe you. Other than that I don't know how you'd be able to prove you're you and not have to change your name in court or switch it on Facebook. :/ I do know a friend of mine on Facebook who has always used his preferred name etc. and nobody got onto him from Facebook or anything. I'm not sure why they'd target you or even know it was "the wrong name" or whatever the issue is.
I'm sorry you have to go though this either way and I hope it all works out for you!