A hash isn't really the right construct for that purpose. If Blizzard just produced a salted hash of user_id & realm, it would be pretty expensive for them to reverse it, even given possession of a complete list of user ids and realms. If they didn't salt it, they could keep one big "rainbow table", but then again so could anyone else.
What you'd really want is encryption. If you chose symmetric encryption, the key would be in every client (easy to steal). If you chose asymmetric encryption, the message would suddenly get much, much bigger to the point where it's harder to reliably encode in a screenshot.