Can my iPad and iPhone share iTunes, but have different game centers? Trying to merge
Yes! Just go to Settings - GameCenter and you can sign in using the applicable Apple ID you want to use for that service.

You can use a different Apple ID for each individual Apple service (iTunes/App Store, iCloud, iMessage, FaceTime, GameCenter)–you just need to go to their settings page and sign in with the one you want.

Do keep in mind that some games use iCloud to store their game data, which may affect your plan.