You definitely don't need to prove anything to get an Oyster card. Be sure that if you do prepaid instead of the weekly that you can not use it on the overground or the tram.
I live in Wimbledon and the lines can be long, but there is more than one window that does Oyster. In fact, on Sunday, only one window didn't do Oyster. And I got my card at 8:30 on a Sunday night, so it wasn't too early. I think they are trying to improve!