I think this is analogues for the Woodward effect, but macroscopic:
We assume a spacecraft consisting of a broomstick, a donut and lots of gear for storing and transfering mechanical energy. Take the donut, spin it till the outer edge moves at relativistic speed (We assume a tough glazing), and push it along the broomstick, brake it's spin and then the donut, move donut forward along broomstick, repeat. Since the donut is heavier when it's being accelerated backwards then when it's braked/accelerated forward, there should be a net momentum. Since nothing leaves the broomstick, this has to be wrong. But where?