The tides are already transferring energy from the Earth to the Moon, making it go higher over time.

I suspect that adding more drag to the tides could either boost this process or reduce it, depending on the timing of the energy harvesting.

What about the compromise, the Moon in geostationary orbit: would be great for broadcasters (no more broadcast satellites, just plant antennas on the moon).

The energy in tidal currents comes both from the moon orbiting the Earth, and from the Earth orbiting the sun. (Technically other heavenly bodies also have some effect, but it's negligible.)

I've always wanted to have a bigger Moon, this might be a way to achieve that.