[Gravityloss]

Interesting idea.

You could carve a few cubic kilometers of rock. Raise it when the moon is at zenith. Lower it when the moon is on the other side of earth.

There was this idea to use a carved rock cylinder for pumped water energy storage (have the water under the cylinder).

You could even combine these.

Of course, in reality the difference in gravity is so tiny, the round trip efficiency is too low, killing the idea.