And then the wheel is instantly out of balance, causing the device to recoil in the opposite direction of the released projectile.
Voila! The laws of physics are obeyed!
But it does not recoil in that direction. The release of force is perpendicular to the direction the projectile leaves. The force the projectile is applying to the gun barrel is very nearly perpendicular to the motion of the projectile at any given moment. At release the barrel suddenly stops receiving that outward force. The projectile continues moving in the direction it was moving in at the moment of release and the rotating barrel suddenly (instantly) experiences a lack of force in the radial direction. Imagine you are holding two buckets of water and spinning around. The moment you release one you are unbalanced and the remain bucket causes you to wobble in a circle. effectively your center of rotation was moved very suddenly. The actual acceleration of the buckets was absorbed long ago by your feet pushing in circles on the ground.