Sorry, Batty. When you ask a bunch of strangers on the internet for advice some of that advice will surely be crap. That said, I read a lot of "I" in your OP. This in particular: "I'm shutting him down for a month or so..." -- sounds like a heavily invested dad micromanaging his kid's baseball career at an age when the kid should be taking responsibility for his own choices and training. But that's just MHO so please feel free to disregard, and best of luck to both you and your son.
Disagree, the kids a So in HS not college. He probably doesn't drive yet, so dad needs to be involved. We r not talking about taking to coach here. Same as parent worrying about kid academically and wondering if they should change tutors.
Imo tee work should never stop. Son needs to get pop up net in basement/garage and hit all the time, practicing what HC taught him. If he's been successful with HC so far, I would not change just yet. Work on basics over winter and see how it goes. son is still young. They fall back into bad habits. Just because he was doing well doesn't mean he won't have slump, happens to everyone.
Glad you brought up mental part of hitting. Normal for them to get down on themselves. I would recommend book-The Mental Game of Baseball, by H.A. Dorfman. A good read.
good luck