My guess is that tidal forces (speeding up Earth's rotation, but slowing down the moon (the opposite of what happens in this reality)) would dominate over atmos

My guess is that tidal forces (speeding up Earth's rotation, but slowing down the moon (the opposite of what happens in this reality)) would dominate over atmospheric drag. How much atmospheric drag affects a spacecraft in LEO is a complex problem, being a function of the spacecraft's mass, surface area, orientation, etc. However the moon is dense enough that it would probably be fairly content to barrel through the sparse atmosphere for quite some time.