When they first ran into the tuck under problem with the P-38 it was concluded that it was caused by the airfoil reaching Mcr, i e small regions of supersonic flow formed on the upper surface of the wing. The corresponding shockwaves made the airflow separate far ahead of where it normally would, causing a drastical reduction of lift and downwash. And, as you said, this gave the elevator less negative AoA and more (positive) lift, causing the plane to pitch down even more. They eventually solved it by adding flaps IIRC.
As for how a stall affect the location of the center of pressure, I really should know but I don't. It should move forward in a normal stall as you lose lift on the rearward portion of the upper surface first. I guess it is entirely possible that the opposite happens in a tuck-under situation with the airflow separated further ahead on the airfoil. If I'm wrong I'm sure someone will correct me here.
Not sure what you mean by "root spoilers". Simply spoilers mounted near the root of the wing?
Cheers,
/ft