Kudos on acknowledging this and addressing this. Anyone with much experience in the hunt field is familiar with riders who seem to have no awareness that their horse is behaving dangerously.
Given what you described of your horse’s behavior (and in particular that it is getting worse rather than better with experience), I would consider the strong possibility that he is getting pretty mentally stressed by hunting. This does not mean that he won’t be (with time and taking it slowly) a good field hunter. But right now, he needs for things to slow down so he can mentally “breathe” and decompress.
Every horse reacts to hunting differently, and there is no training rule that says a horse has to go out regularly or for the full season, especially in his first season. My now retired field hunter (who became terrific first flight horse) intentionally spent his first year out cubbing and then hunting with the hilltoppers, and then we stopped mid-season with him doing very well. I had no intention of moving him up at first, as I wanted him to become fully comfortable with a slower pace before asking for him (mentally) to handle more.
Even though the physical aspect of foxhunting can be challenging, from the horse’s perspective, my impression is the the mental aspect is significantly more so. And in training them to be a good field hunter, we need to be really aware and ready to adjust the program in response to how the horse is handling things mentally.
If your horse’s behavior is a manifestation of him getting mentally stressed by hunting, in addition to the exercises described in the earlier posts, consider taking a couple of weeks (or even a month) off from hunting. Let him decompress. When you come back to it, consider staying in the hilltoppers until he is fully comfortable with that. (Given that he is a OTTB, you ride him regularly, and that you would be with the hilltoppers, he still would be plenty fit. And that little temporary loss of foxhunting fitness may work to your advantage.)
I learned to foxhunt from a husband and wife team who trained horses to be excellent field hunters. Typically these horses where athletically very talented but had developed serious behavioral problems from having been mentally fried from being pushed too much too soon . I learned a lot from these trainers, and it really impressed upon me how slowly a horse needs to be brought along, and how aware we need to be of when they are showing signs of mental stress / decompensation (which not uncommonly manifested by them demonstrating dangerous behaviors). It does not mean that we completely back off with the training program, but it does mean that we need to change and adjust the program so it accommodates the needs (physical and mental) of the individual horse.
Hope this is of some help.