Isn't the whole point that it is the warp that is actually moving and not the ship? The warp is not bound by the speed limit, and the ship is not moving so no relativistic effects are experienced.
No, the ship is moving and it isn't. Meaning, the ship's driving in a constant of acceleration creates the soliton wave to begin with. Then as the wave builds (shrinking space ahead, narrowing time zones ahead, squeezing gravity's gridlines, before the ship) the ship will grow in 'surfing', in easier riding, its own creation of the soliton (warp bubble) wave. Think of the accelerating expansion of the universe being locally caught up to and reversed to a localized accelerating contraction of the local universe. The power surf, the easier and every easier ride because the soliton (warp bubble) wave created, is doing more and ever more of the work . . . thus the ship needs less energy instead of more to keep accelerating (to keep riding the surf in the shrunken local distances of the universe ahead).
I write and keep seeing an aircraft powering to reach altitude, then having reached cruising altitude and velocity pulling back on power, easing up on acceleration somewhat, so to cruise . . . so to surf a wave. So it will be with constant power created soliton waves dealing in gravity's flexible gridlines narrowing light's coordinate points past-future time zones of SPACETIME.