Is it possible to balance a tail heavy plane with a vertical prop on its tail?
Tryin' to cheat the laws of stability, aren't we?

In both videos it is evident that the plane is unstable in pitch. Adding a lifting prop will not change this, because changes in the angle of attack will not affect this prop's lift force much. Stability is achieved by shifting the center of gravity forward, so that changes in the angle of attack will produce additional forces which create a counteracting pitch moment.

In fact, adding the motor will shift the center of gravity even further back. In some cases this helped because the increased pitch inertia made it easier for the pilot to keep the aircraft under control.

If you need to keep the center of gravity so far backward, add artificial stability with a pitch rate sensor and a microcontroller which deflects the elevator appropriately. There are many pages with Arduino-controlled artificial stability solutions - take your pick!