Depending on the situation that could cost a ton of money. Not only would you have to buy a new copy of WoW (vanilla) but you'd have to buy a copy of BC, WotLK and soon Cataclysm. Depending on where you buy the older games from you could be looking at nearly $80.00 (not including Cataclysm). On top of that, you'd have to spend several weeks leveling up your character, not including losing out on any achievements you might have collected since 2.4 was released (like the 310% drake from "Glory of the Hero").
Might be a pain in the O.O to give them the info they're asking for... but I'd rather do all that then have to redo everything I've obtained since WotLK came out.
Also, you might want to look into just giving cust. serv. a call, being able to talk to someone on the phone might work out a little quicker (specially if they can put you on hold and ask a manager about the issue while you wait).
As for the real ID system I think you're mistaken about how that actually works. There hasn't been anything verified yet that someone can get your real ID information from you through an addon since they did patch 3.3.5a (I'd have to find the blue post that confirms this). The only way someone can see your real name is if you give them your email address attached to your
Battle.net account, and you should only do that if you trust the person your giving the address out to.
Blizzard has already said that they're removing some of the planned Real ID uses that they were talking about for the launch of Cataclysm, one that even made it to national news (in the USA) was using real id as your "screen name" for posting on WoW, Starcraft or Blizzard forums. The people cried foul and Blizzard said 'whoa... ok so we wont do that I guess'.
Good luck with the name change though.