Q:

Which perturbs LEO satellites more, the Moon's gravity directly, or the changes in Earth's gravity field due to the Moon?

Reading this tutorial answer on models of Earth's gravity field I was realized that I'd never thought of it as time dependent. According to it, the biggest short term changes in the shape of Earth's gravity field in LEO space is the movement of the tides.
It got me wondering, if that question were relaxed and it asked about the total gravity field of the Earth-Moon system, what gravitational force would tend to perturb LEO satellites more, the modulation of Earth's gravity field due to the movement of the tides, or the gravity field of the Moon (which is also causing those tides)?

A:

The Moon's gravitational field itself, by far. The perturbations on a satellite due to the reshaping of the Earth due to the Moon and Sun's gravity fields are small effects.
The primary impact of this secondary tidal effects is a slight change in $J_2$ ($C_{2,0}$). A description of how the Earth tides affect the gravitational field is in section 6.2 of IERS Technical Note 36. The next section (6.3) covers the impacts of ocean tides. These are smaller yet.

