Southwest is basically asking for proof that you are the same person and not two different people trying to transfer an account from one person to the other. I don't know how you can accomplish this without old and new identification. It's the same process as someone who gets married and changes their last name.
A different idea would be to just create a new account with the correct information. Now this isn't the really an option if you have A-List status or a Companion Pass as you would lose access to them. If you have points accrued under the nickname account you can still use them to purchase flights for you under the "new" name, but no new points would go into that account. Just an idea, it depends a lot on your current situation. If you're an infrequent flyer there's probably very little downside to doing this.
--TheMiddleSeat
