How do they balance the spinning mechanism at those
high RPMs once the rocket is released.
Just don’t see this as workable.
They might be able to release the counterbalance at the same instant that the rocket is released. That would mean two exit ports, one for the rocket and one for the counterbalance. I don't know how they would handle the tremendous momentum of the counterbalance. Maybe use a water-containing cylinder that would break up when it hit the ground?
Depending on the mass of the thrower, natural freqeuncies, etc., could be managed.