No one here can promise what app review will/won't accept/reject. App Review isn't here, and there are no pre-reviews.
My opinion is that you'll need to run the app review gauntlet to get an 'answer' that even smells definitive.
I don't believe any licensing is required. The use must be consistent with the intent of the trademark - the Game Center icon must send you to the Apple Game Center. When using other Apple Trademarks you must indicate the trademark with an ®. I'd add that to the icon.
You can ask them here:
FWIW, we wound up not using the logo due to consistency with the app settings screen (where there's a Game Center achievement reset button but the regular GC logo color scheme looked inconsistent), so I never found out whether it would have passed review or not. We did write to legal, but we're probably too small for them to worry about and they never responded.