Ignoring the concept of "can we apply that much delta-V to a planet?", I'd be interested to know whether it's believed that there exists a "Goldilocks zone" suitable for life at all stages of a star's life. Intuitively it seems like there should be, I'm not sure.

Of course, it should be pointed out that the common understanding of "when the sun becomes a red giant" may be a bit flawed; the sun will cool and expand, then collapse. On a human time scale, it will spend a lot of that time as a red giant, but if you simply took the Earth when its orbit started to be crowded by the inner edge of the Goldilocks zone and put it in a new orbit, that new orbit wouldn't be anywhere close to an eternally safe one. Indeed, I suspect that the outermost of the orbits required for the giant-stage sun would be too far from the sun at the time we'd first need to move the Earth.