OMG!!!! To PDUBBER:
Thank you Thank you Thank you Thank you Thank you Thank you Thank you Thank you Thank you Thank you Thank you Thank you Thank you Thank you Thank you Thank you Thank you Thank you !!!!!
Your solution worked like a charm! I had absolutely no idea that the Apple ID field can accept a Game Center ID that works with your Apple password. This solution even works on iOS 7.1.1. After all that hair pulling I did, all I had to do was enter my alternate Game Center ID in place of my Apple ID and enter my password. So simple, and yet it took me hours to find this answer. I can't believe it was that easy.
Funny how nobody can get an answer to this problem from Apple but you found the answer and posted it here. In fact, I never see anything but users on this forum. It's like Apple Tech Support people never EVER come here. This is just a place where users go to solve their own problems themselves.
Anyway, thanks a billion for this. May you grow very old and very rich!