OK, so I still don't get the details, etc, but I tried working out an example that was throwing me off.
The complex background doesn't matter, but for completeness: we leave home and get up to 0.85c. We slow to a stop, somehow. Weve traveled about 1000ly. In the unlikely event that I understand correctly, and that our mass is negligible (it probably isn't), it's been roughly 1160 years from home's PoV, and 580 from ours. At our destination, we build an FTL engine that can get us up to 100c, and turn around to go home—which, from what we can see at our destination, appears as it was 160 years after we left.
After going in circles with what to do next, I realized I should just try to work out the math. At 100c, the Newtonian interpretation would be that it takes us 10 years to make the return trip. This might be what the homeworld observes, though we'd probably appear to have imaginary length/mass if they could see us.
But that 0.85c  contracts time by about 0.5 thing? That tells me that sqrt(1-v^2/c^2) is our time dilation. Plug in 100c for v, and you get sqrt(1-10000). Let's round that to ... 100i. Umm.
So the amount of time that passes is 1000 imaginary years?
OK, maybe our ship is doing <mumblemumblebabble> to trick the universe into thinking we have imaginary mass. I'm going to ignore the crap out of the headache that is Tachions, and assume this applies to the energy around us or some crap, whatever.
So either we have a mass of i, and therefore experience 1000 years (exactly the same as if we were traveling at c under Newtonian physics), or we have mass/energy of -i, and experience -1000 years, which I don't even the what that's not time-travel how can you observe time reversing when your neurons are part of the reversed time do you just become antimatter or something?
This renders FTL meaningful for our homeworld, who should see us return ... uh, either 1170 or 170 years after we left? But I think the result for most velocities > c (or enough greater that the 1 is negligible), it's fairly useless for the FTL traveler. You might have to go FTL to experience traveling at light speed.
Should try a small number, just to check. At 2c, we get sqrt(-3). If we have m=i, that means we experience over 1.7 times the duration a stationary observer would observe. This will asymtotically approach v/c as v increases. Since stationary observers would see something like 1/v times the distance, the v all but cancels.
That is simultaneously hilarious and depressing. But if you need a hyperbolic time chamber, just travel at 365c?

If m=-i, then we should arrive 160 years after we left, but what in the world do we experience during this trip? Seriously, are we turning into tachionic antimatter? What does that even mean?
But then we can catch up with our past self. Not only that, but the light from our arrival will catch up to them, because c is constant in all reference frames. Since past-us would be about 120-130ly away at the time, and experienced 80 years, 200 years into the trip, we should have seen ourselves arrive at home. Do not like.
We could construct a path so that information from our future conveniently never catches up with us, but neither attempt at handwaving away i is helping with the general case.

Just switching to wormholes whose distance/transit times are larger than any relativistic age differences between the mouths is quite tempting. Mais non; cette n'est pas possible pour tout mes problemes.

I think the only way this works is if the universe has an error-handling routine for anything involving the square roots of negatives, and it just happens to be maximally convenient for fiction. That is ... rather unsatisfying, but here we are.