IMO as a teacher the correct action would be to try and resolve the issue of cheating at its core. I feel that staying in the new gc and waiting to see if students cheat again is a lack of integrity - he is waiting to see if students cheat rather than teach and grow them.
Dude is not going to solve "cheating at its core". Read some of the shit going on in this thread even. He's not going to solve the "I am a proud cheater because not cheating means you're stupid" guy. He's not going to solve the "I need this piece of paper because companies are dumb and require a piece of paper instead of proof of competence" guy.
You still haven't really shown how this is a problem with his integrity, you're just using the word in a sentence.