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)?