I personally am not a fan of doing serious engine mods such as boring unless you are a serious racer, it just opens up too many cans of worms and can cause problems such as this. I don't think the got cam should be causing you any grief, should just make the valves open and shut quicker at the best time to give optimum combustion. I have been trying to find a way to blame valves or clearance problems but right now I can't think of any.
When dad had trouble with the skidoo and the over-size piston it started with the same problems your experiencing, would start perfect when cold, would work while on but if shut down would be a real pain to restart. Usually it would help to hold the throttle in a bit to give her that extra pop to get her goin and after that she would work till she was shut down again. It eventually came to the point I previously described where she shut down on her own.
I don't know what to say about the tranny, that's a weird one for sure. Sounds like it's next to destroyed. I'm seeing why this quad was gottin rid of, hope you got a real good deal.
Kenda Front Max Tires
Maxxis RAZR Tires
K&N Air Filter
Power Madd Hand Guards
250 Moto 4 (Under Construction)