[tankenmate]

Balance scales work independently of gravity, both sides of the scale are equally affected.

[stickfigure]

Only if the moon is directly overhead.

[tankenmate]

Down to eight significant figures it is an issue (effect of the moon's gravity on the surface of the earth peaks at 1.1 × 10−7 g), but you can always point the normal of the axial plane of the balance in the earth relative right ascension direction of the moon to avoid this effect (regardless of the moon's relative declension, δ).