"If you could, either relativity is wrong, or you can use this to make a time machine to access the past."
Well, relativity is most certainly 'wrong', in the sense that there is more to the universe that it does not cover.
Saying "there are things it doesn't cover" is not the same as saying "it's wrong."
Of course, that's not really wrong, any more than Newton was wrong. Newton was not attempting to model relativistic effects, and Einstein was not attempting to model string theory. And luckily, we are not made out of light, so the speed of light has little to do with
The phrase "speed of light" is historical usage. An equally accurate phrase would be the universal conversion factor from units of space (meters) to units of time (seconds). It doesn't apply just to light, it is a universal constant that applies to pretty much everything in the universe, not just light. Most particularly, it applies to gravity.
a machine designed to alter a local region's gravitational constant,
If you alter the gravitational potential, the gravitational time dilation will mean that external observers will observe your speed of light as having changed. If you're in a gravity well, your clocks have slowed down, so it looks (from the outside) as if light has slowed down. But, inside the region of altered gravitational potential, if you measure the speed of light, it's still moving at the speed of light. You can't travel faster than the speed of light where you are.
that can travel at arbitrary speeds relative to the destination because it carries it's own engine with it. Relativity does not forbid a starship.
If you could travel at arbitrary speed (in any direction, in any reference frame), relativity says you could make a time machine. Space is time. If you can travel faster than the speed of light you can travel backwards or forward in time.
But none of that would ever allow you to travel back in time. That would be akin to traveling in a negative direction.
Right. If you can travel faster than the speed of light, but only one direction, then you can't make a time machine.