Would the travel in this case be linear? I would expect slight deviation in regards to curvature...

I would say non-linear. We may indeed follow the path taken by light.

Note: Time travel could possibly utilize another form of phenomena other than just the sol, correct? I've always been fascinated with the ability of perhaps warping space-time with some kind of distortion apparatus, along with a GPS peripheral to pick & choose your position in the cosmos :-)

Indeed we need another way to travel. Otherwise, as I said, we need an infinite amount of energy to accelerate the ship. Warp drives are for the moment hypothetical. They may or may not be ever built. But it is cool to dream about options :D