It's a fascinating idea from science fiction perspective, but moving the Moon anywhere will probably have devastating effects on the Earth's biosphere
A more likely scenario is that the Earth's orbit will move outwards as the Sun expands, since stars in the red giant phase(s) lose a lot of mass. It's plausible that the Earth/Moon will not be destroyed, physically. But in that case all life would have been obliterated due to the immense temperatures associated with having the surface of the Sun within a few million miles (or less) of Earth.