Were my calculation wrong?
Must have been! Distance, as you say, is very nearly the same, and the Moon's mass is 0.0123 that of the Earth. So Sun-Moon attraction will be 0.0123 that of Sun-Earth.
As far as "cocktail-napkin" calculations, try this:
Earth-Moon attraction = 1 X 0.0123/1
2 = 0.0123
Earth-Sun attraction = 1 X 333,000/390
2 = 2.18
Moon-Sun = .0123 X 2.18 = 0.0269
Edited by Coragyps, : add calculations