iOS App Review: no Game Center features within the app (x-post from S.O.)

(I just posted this on StackOverflow, and was advised I might get better results here, so here goes...)

I've developed an iOS game - a simple sidescroller shooter. The game uses GameCenter for leaderboards and achievements. In my own build, in Sandbox mode, the game authenticates, welcomes the user, displays the leaderboard overlay, everything works great. In the game's menu is a button that connects to Game Center if it's not already authenticated, then pulls up the Leaderboard overlay. This works great in Sandbox.

I submitted the game to App Review, and it was rejected on the basis that the button mentioned above in fact does nothing. If I try to authenticate without Sandbox mode, I get the message in Xcode that my application is not recognized by Game Center. Based on this thread: iOS 8 beta 5 Game Center Sandbox wont recognize my app, I assume this is expected behavior for an Ad Hoc build.

My problem - since the submission build is signed with a separate profile and certificate, I will never have a way to see exactly what the reviewer is seeing with Game Center if I understand everything correctly. In iTunesConnect, everything is and has been turned on the whole time - Game Center, Leaderboards, and Achievements, for both the app and for this version of the app. Inside Xcode, the build does indeed have GameKit enabled.

I've posted about this to Apple's forums (both to Game Center and to App Review) with no response. Since the game was built with Unity, I also posted to Unity's forums for the heck of it, though I doubt this is a Unity thing. No response. App Review says to go to Developer Tech Support (DTS), and DTS says to talk to App Review.

Am I doing something wrong? Do I need to actually create an iTC Sandbox user for the reviewer? (This would still require them to use Sandbox mode, of course). Do I appeal the review? Is this not simply a case of them needing to "turn on" GameCenter for the live version? All I see on the Internet are repeats of the "just use Sandbox mode" suggestion, which does indeed work for development&distribution but not for submission/review.

Any help is GREATLY appreciated, as I'm getting a bit frustrated with the idea that nobody's seen this happen before.