Get the game center application back on the MacBook Air

The simplest way is to delete the game from your iPad, which will destroy any local stored game data, and will prompt you to remove the app from Game Centre (which I assume you want to do if it's corrupt on the Server). Then redownload the app (or re-add it in iTunes in your computer). This should allow you to start with a clean slate.

That login banner is a provided aspect of the Game Center API in the iOS SDKs, as far as I'm aware there is no way as a developer to disable it from opening when the Game Center login completes. As a user, there is certainly no way to disable it, because it is part of the application's compiled source code.

However, note that it's not the banner that is causing the delays and skips. It is actually the entire login process to Game Center. My only advice is when you open a game that has Game Center logins, open it up to the first functional screen, and just wait the ~5-10 seconds it takes to reveal the Game Center banner. After the banner has displayed and disappeared, proceed normally and you should not be affected any further.