How to recover a game center account where I forgot ID and password

If I understand your question correctly you can:

• sign into your our Game Center on your iPad 3
• delete the app from the original iPad (making a backup first)
• sign into your friend's Game Center on your iPad
• reload the game and verify the correct village loaded

Once that is done, you should have both villages intact.

How open are you to creating a new Game Center account?

If yes, then follow the steps below:
Steps are taken from http://support.apple.com/kb/HT4314?viewlocale=en_US&locale=en_US

1. Open the Game Center app
2. Tap on your email/username and click sign out
3. Tap on the Create a new account button
4. Follow the steps on the screen
5. Sign in to your new GC account and open Clash of Clans
6. Congrats! Your village should be linked to the new GC account. To test it out, sign in to another device with Clash of Clans and tap on OK/Confirm

If no, well..... * shrug *. IDK what to do.