I'm not sure I follow; if the object is held rigidly to the launch arm until the point of release, it's going to have a lot of angular velocity, sure, but that doesn't imply rotation, does it?
But see my other response above -- this does not seem like one of the bigger issues when the centrifuge arm is much longer than the payload. And we already know this launch system will only ever work for small, non-human, durable payloads.
But isn't the payload rotating around its own center of mass because the centripetal force acting on the "front" of the payload is not parallel with the centripetal force acting on the "rear" of the payload? As soon as you cease to apply the centripetal force (i.e. release the projectile) you're no longer going to generate any torque.
Imagine the payload being held by ropes on the back and on the front and then both ropes releasing exactly as its center of mass passes through horizontal.
Attach the payload at two points in a narrow V, equidistant from the center of mass. In the instant before release, fire pistons that slightly extend the upper arm and retract the lower arm, applying the exact amount of force you pre-calculated was needed to counteract the differences in rotation between those two points. Release while this (hopefully uniform) force is still being applied.
Yeah, you're jolting the hell out of your payload, but that's peanuts compared to what it's about to experience anyway.