Good news - you can remove Game Center access for a specific game (like COD) on one iPad without losing your game profile or progress. Here's how:
Important first: Your COD game progress is saved to your Activision account, NOT solely to Game Center. As long as you're signed into your Activision account in the game, your profile and progress are safe in the cloud.
To remove COD's Game Center association on the iPad you want to free up:
1. Open Settings > Game Center on that iPad
2. Scroll down to find the list of games that use Game Center
3. Tap on Call of Duty and you should see an option to remove or disconnect it from Game Center on that device
Alternatively:
1. Delete the COD app from that iPad (this removes local Game Center data for COD on that device)
2. Your Activision profile/progress remains intact in the cloud
3. On your other iPad, sign into the COD app with your Activision account to restore full access
Note: Game Center profiles are tied to your Apple ID. If you have two Apple IDs and want to sign into COD with a different one on the second iPad, sign out of Game Center on that device first via Settings > Game Center > sign out, then sign back in with the preferred Apple ID before launching COD.
Your hard-earned COD progress will not be lost as long as it's linked to an Activision account.