I think the mass increase is not supposed to come from the forward/backward motion. Rather, the mass increase comes from rotating the ring. Imagine the ring at

I think the mass increase is not supposed to come from the forward/backward motion. Rather, the mass increase comes from rotating the ring. Imagine the ring at the front of the ship spinning up to relativistic speed, then you push hard on it to get your ship moving a bit. Then you slow the ring down again before it hits the back, so it's much easier to bring back to the front.