I wonder if lunar space elevators might be the fix here. If I understand correctly, such an elevator would not be as subject to the perturbations since the tension would keep it's orbit stable (is it still an orbit if it's tethered?).

Another option might be a LORAN style system put up on towers. With lower gravity and no atmosphere I imagine we could stick transmitters up very high without super complex construction, maybe even just a giant carbon fiber tube with a transmitter at the top.