If the plane went from the tail providing a down force to providing an up force, the required "pull" would no longer be required, and there would be a control force reversal, neither of which are certifiable.
If the CP is right on the CG at a certain angle of attack, but the fixed horizontal tail is set to produce a nose down moment (lift) at that same angle of attack, in level flight the elevator must be deflected upward by the pilot to balance and overcome the fixed horizontal tail's force. That means he is pulling back on the stick.
Then if the angle of attack increases from there, the fixed tail lift increases, over-balancing the developing nose-up CP shift on the wing (by design). So then you have to add still more back elevator input... which is by definition static pitch stability.
I can see this becoming complicated in design, assuring static stability over the whole CG range. I think you have to make sure that at the angle of attack where the CP has moved forward to the CG, the fixed tail has already transitioned to lift.
God help us if we have to go through the mental exercise for a stabilator
