JPL HORIZONS features orbits interpolated from actual data rather than pure simulations. I suspect that what happened here is that there is no orbital data for the three sections of data that don't exhibit the oscillation. We know for a fact that there is no raw data for one of those sections because it is in the future. I spoke to a controls expert and he said my suspicion is probably correct.
When you use the webtool it says there are several data series with all but the first having the word predict at the end of the line. The dates at the end of those data series correspond to the returned oscillation of the data. I suspect that a new data series is begun when contact with the spacecraft resumes.
I believe the oscillation period of ~14 days corresponds to half of the moon's orbital period. Gravitational perturbation by the Earth would then be the main driver of this behavior. The fact that these oscillations occur quite visibly in the absence of real data indicates that the model is doing a decent job of accounting for this perturbation. As for why the amplitude drops considerably, I can't say for sure since I don't know enough about the model used, but I can guess that it's related to the unaccounted for nonuniform density of the moon and non-gravitational perturbations related to its orientation relative to the sun, such as light pressure and outgassing.
If you look at the eccentricity data you will see an oscillation with a period of ~14 days on top of an oscillation with a period of ~27 days, with some higher harmonics scattered. There is also a strong ~27 day period oscillation in the periapsis and apoapsis data. It might be interesting to apply a fourier transform to all of this. You could look at the relative magnitudes of perturbation by period and perhaps try to isolate some behavior with a period of one year.