I agree, there has to be a recoil, if something shoots out the front, and nothing shoots out the back, conservation of momentum demands that there be a recoil.
I am studying physics in school, and I think that you are forgetting centripetal force, the reaction force happens before the projectile leaves the weapon. That force is the projectile pulling out away from the center of the centrifuge and when the projectile leaves the weapon the force goes with it. Have you ever spun around with a heavy object? the object pulls on your arms and if you release it it goes flying without pushing you back. But that doesn't mean that I believe that the weapon works as they say or even works at all there are many things that make me say it wouldn't work.