I believe I have a fool-proof solution for anyone seriously interested in FTL travel, also realizing they probably have read this already. I must assume me and you inhabit different universes, though, but with logic being omniversal and my semantics understandable to you.
Since infinite energy (please clarify if you understand the concept of energy though) is then something possible and vital to acquire there. I believe following your logic a sort of pseudo-perpetual motion machine without a meaningless time component would do the trick. If infinite energy is available without said great machine, I'm sure you'll find some easier way to acquire it in nature, and found already.
Since FTL travel being possible would mean causation is nothing to be concerned about, (not sure if it ever existed with the wild absence of time as I know time) you already have probably invented the perpetual motion machine or long ago too.
And even if you didn't, the notions of causation and time being pretty useless in your universe, you can probably already use all that delicious infinite energy of exotic kind which only took infinite time aka 0 time to pop out as a sideproduct of that perpetual motion machine.
Now you have and always have had something to travel the space fast(???) for great win, which weirdly sounds to be about the same as it is here, if time is left out of the equation. (Must consult a certain physicist working on quantum gravity on this concept of space)
...wait, you telling me speed or acceleration mean something there? Disregard all this, I have no idea what kind of reality you inhabit, even my logic doesn't apply. Seems like even string theory has no unempiristic clue about the real weirdness of the multiverse... I think we need to rewrite all our science.