For game messaging, maybe the best solution would be to have a 'messages on/off' toggle button right on the game page, around the area where the move button is, or under it. That way if someone in a particular game is bothering you with messages, just turn them off. Next time they try and send a message, they would get a little notice that you have the message option off. You could just turn it on or off whenever you wish.
As you say, this really isn't the moderator's job. Their function is to moderate the public forums, not game messages or private messages.