Jamie, have you looked into the concept of "Force Fetch" yet? It is predominantly a piece of the training process for Retriever breeds, but I don't see why it wouldn't also work for treeing dogs. Any dog that with retrieve can be force fetched.
The entire process of force fetch can be relatively long and tedious (weeks) but the first portion of it, teaching the "hold" command, could be very useful for your dog(s). One could argue that the process should be called force hold instead of force fetch... But that's beside the point. I'm going to make this incredibly brief, but basically you out the dog up on a platform or table, away from other distractions and remove the ability to get away. You start by putting an inanimate object in the dog's mouth and teach the word "hold." Put it in the mouth, say "hold", good dog. Etc. Teach that word. Once they know it, you start to incorporate pressure. An ear pinch while holding the collar with pinky and ring finger works well. Pinch and hold ear, say "hold", when dog reaches for object and holds, release pressure from the ear. Essentially you're teaching an "OFF'' switch to the dog... Do as I say and the pressure goes away. If the dog tries to spit the object before you tell him to, you're ready with a swift well-timed ear pinch... "No"... "Hold." The eventual goal is the dog grabs something when you say hold (eventually transitioning to the word "fetch") and doesn't release the object until you say so... No exceptions.
You're probably wondering, well what the hell does this have to do with a hard mouth? It's an indirect benefit of the process of done correctly. While training hold or force fetch, you should also insist the dog develops good hold habits... I.e. not rolling the object around, not chomping down, not playing with it, etc. This is where it can get tricky. Some dogs, especially those that have bad habits to start with, can have difficulty grasping this concept. My dog was that way. Start by using a GLOVED hand as the object. Put your hand in the dog's mouth. You can feel whether the dog is actually holding it or if it's just simply allowing to put something in it's mouth. There is a difference. Once you establish the corrections and the dog knows whether it's fucking up or not, you can move on to other types of objects. I particularly like objects that aren't balanced... Like a hammer. A dog cannot roll a hammer around in it's mouth without dropping it, due to the heavy weight at one end. That have to actually HOLD it by the handle in a consistent manner in order to avoid getting the ear pinch. Plain and simple.
I know I'm getting long winded now so I'll wrap this up. You can google force fetch for days... Literally. I think it's worth your time to look into it. You have to slowly teach this concept to the dog. It won't happen in a day or likely in a week. But once it knows what the force fetch process is all about, you then have a TOOL available for when he slips up in the field and starts munching on a squirrel. Set him on a log and start going through FF with that damn squirrel... He will eventually understand that his job is to grab it and bring it back to you, and hold it until you say not to, and that it doesn't involve ANYTHING other than that extra... No playing, chewing, eating, whatever.
Anyway, sorry for the ramble. Those were just the footnotes, so fire away if you have any questions.