Tensioned Suspension – Sculptures Suspending Themselves

I suspect the difference in gravity between when the Moon is overhead (or at the nadir) and when it is sideways would be much less than the effect of the constant small tremors that would be felt if the object was set on solid ground.

Maybe a kind of low-pass system would allow it to react to slower change and ignore the higher amplitude, higher frequency of the tremors， though.

Probably not. The moon has a mass of about 7.34e22 kg, the minimum distance from the moon to the surface of the earth is about 363,104 km, or 363,104,000 meters, and the constant of proportionality for gravity is G = 6.673e-11 N m^2/kg^2, so the attraction between the Earth and a 1kg object on the surface of the Earth is 3.7e-5, or about 8.3e-6 pounds force. That's not much to work with.

(Did I do that right?)

You could, but you would probably have to use a timing mechanism that is synced with the moons cycles rather than the moons gravity itself.