David: Try that black hole example. Your nominal terminal velocity at the horizon equals lightspeed, and deeper in, it's more than lightspeed. It can be entirely legal to travel faster than background speed of light, and even the local speed of light, as long as you aren't exceeding the local velocity of light, in the direction of motion. You just probably won't be able to report your success back to any observers outside the hole.
That's the basis of the Krasnikov tube idea. You activate the tube, ride the artificial gravitational gradient to your destination arbitrariy fast, then, if the tube polarity is reversed, ride it back again, arbitrarily fast. The way your signals get mangled means that round-trip SR-style definitions of dates and times go all to hell, but those definitions also break down in some pretty mundane everyday situations, and as long as you don't actually arrive before you left (and why would you), there's no underlying causality paradox. The optics get scrambled, but that's about it.
Certainly, the SR definitions go a bit mental in this scenario, but they also go a bit mental in the presence of conventional gravitational fields, and we don't say that therefore gravitational fields can't exist
So trying to disprove the existence of warp drives using special relativity is a bit crazy. Trying to disprove "metric engineering" solutions by using a theory that presupposes flat spacetime is like trying to disprove the viability of aerofoil-based heavier-than-air aircraft designs by presupposing the absence of air. One can certainly obtain a rigorous disproof, but the disproof is pretty much worthless, because it's based on the simplifying assumed absence of the very effects that are required to make the hypothetical mechanism work (in this case, gravitational distortion).
This doesn't necessarily mean that we really can build a practical warp drive – there may be other insuperable obstacles – but the usual reasons given for why we can't do it are