What would a determinist have against the idea that time travel is impossible, based solely on Determinism? If something can cause time travel to be possible, surely that wouldn’t violate Determinism.
Logic, however, presents issues - at least with time travel back to the past. If you change the past, you change the conditions that lead up to your time travel back into that past, changing the exact manner of your time travel into the past, further changing the conditions that lead up to your time travel back into the past - and so on. An infinite feedback loop of timelines are required to cover all of these permutations, which begs the question of how you create an infinite number of universes where these alternative timelines occur. If infinite timelines in infinite parallel universes already exist, there are immediate problems in proving they do, and if they did, statistically it’s certain that infinite people from other universe timelines have already travelled to our timeline in our universe an infinite number of times, and continue to do so. If so, they’re all keeping remarkably quiet about it - no matter how compelling some mysterious gadget with magnets and angled wheels sounds(!) I hope the mention of such a thing was not intended to lend any credence to your proposal…
Time is more of a suggestion of entropy than a dimension. Space is just doing its thing and rearranging here and there, with a direction of its progression in rearranging itself only really suggested by states of less entropy “becoming” states with more entropy. To go back in time requires a decrease in entropy, which is problematic.
None of this rules out time travel into the future, but then we’re all doing that anyway - and to do so at faster than the normal rate just requires that you slow down the entropy of your own system relative to the entropy of your environment e.g. cryogenics. You don’t need any sci-fi to do that, except perhaps to unfreeze yourself and live(!)
I think it’s more compelling to think of mass as a dimension than time, since mass curves space itself “into” it like a fourth spatial dimension - but that’s a different topic.