[ombry]

The short answer is that the center of mass will move just as if the weight it static (no rotation). You could balance the plane with the heavier prop and the motor stopped, and then start the engine and acheive the same balance location (provided the thrust force didn't influence the balance - perhaps better to perform this experiment in a vacuum). The flying inverted trim test for CG would produce the same result since this is more or less a steady state test (meaning the flight conditions aren't dynamic through the observation period and thus gyroscopic effects are minimal).

However, this does not mean that the dynamic handling qualities we typically associate with a forward CG will be identical between the rotating and non-rotating mass. A larger rotating mass will induce greater gyroscopic stabilization about the pitch and yaw axes at the expense of greater gyroscopic coupling between the two. This might produce a feeling of being even more nose heavy as the plane might feel less responsive in pitch and yaw.