Next, I take issue with Brett. CG location does directly affect affect stability. Some aircraft flown with CGs at or behind the allowable CG range "porpoise" which is sorta the full size equivalent of stunt "hunting". I have also experienced "hunting" on stunt ships of mine which were happily cured with noseweight. I have also used noseweight and wider line spacing to increase available lift to tighten corners of marginally heavy airplanes.
I won't rise to the bait again. Agreed, that can work, if you cannot adjust the flap/elevator ratio. But I don't see any evidence of it being overweight or stalling here. Simply adding noseweight can cover up the underlying problem, as the limit cycle will likely get slower as you add noseweight, and eventually become a tolerable issue. The problem is that as you add noseweight you also add control load, and slow the controls. Adding spacing will can get the control response back, but it can't do anything about the control loads. That leads down the spiral of having to generate more line tension, which leads to compromising trim to add line tension, which leads to more issues yet.
Hunting in the classic sense IS NOT a simple instability. I have flown airplanes that were tremendously unstable, and had to be reverse-controlled in every corner and even the rounds, but had no tendency to hunt at all. By the same token some of the worst hunting airplanes I have flown were those that someone tried to fix by adding noseweight to the point they just about couldn't do a loop. In some cases moving the CG forward made it worse, or much worse- since, as mentioned above, the leadouts are effectively moved back by adding noseweight, and having the leadouts too far aft is a classic cause (either causes binding in the leadouts or causes non-linear separation effects as the relative wind across the fuselage detaches and reattaches).
In your full-scale example, I would suspect a combination of phugoid effects and either PIO or Auto-PIO. At some point as the CG moves back, the natural frequency of the pitch response slows down, and at some point it will definitely start interacting with the pilot response or autopilot response. A pilot can fix it by forcing himself to go "out of phase", the autopilot just does what it does. Howard could tell us how current autopilot design works, hopefully something better than the crude gain-scheduling systems I have seen. The phugoid poles are weak at best on stunt planes, the PIO it a legitimate possibility but there has to be some non-linearity somewhere.
If you don't have enough lift, adding flap travel WRT the elevator is far more effective than moving the CG forward, and again, doesn't compromise the performance and only hurts the control loading a small amount. Unfortunately, with some designs, the issue causing the stall is not too thin airfoils, inadequate flap travel or area, lack of sealed hinges, but excessively sharp or poorly-shaped LE, which causes it to separate at very low Cls. That cannot be fixed with the CG OR changing the flap travel. And that's about the only case I ever see any more. As long as the engine/prop is running right, anything close to right is sufficient. So I woudn't be too concerned about that.
Fixing the underlying cause is much more effective and will not lead to compromising the performance, which I think is a better solution to look at more likely causes.
Brett