Hmm, the Control Center Remote seems to be built into the iOS, it doesn't depend on the app being installed. I tried installing the app and blocking the bundle ID com.apple.TVRemote which did hide the app, but the Remote was still accessible from the Control Center. I don't see anything else that can block this functionality, Apple would need to add it to their MDM Framework.
We force our Apple TVs into Conference Room mode, so when the Remote first connects to the Apple TV it does prompt for the code on the screen, which can give the teacher a clue something is going on if they happen to see it.. but then the iPad has already been paired. Fortunately we have the Apple TVs configured to prompt for the Restrictions code to get out of Conference Room mode so our students can't mess around.
@chmoi Are these 4th Gen or newer AppleTV's? If so there is an mdm payload called TV Remote that you can play around with to either only allow certain Apple TV's to be paired or only certain iOS devices to pair with them. This will effect both the remote app and the the control centre button.