As far as I remember, the ISS orbits low enough that it requires regular rocket boosts to maintain its orbit against drag from the last bits of the atmosphere. So if the moon orbited there, I'm guessing it would have fallen back to the Earth within a few hundred years or so.
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.
The volume to surface area of the moon is massively greater than the ISS, and only a tiny amount of the surface area is actually exposed to the atmosphere. If the Issue can last for months without a boost, I figure the effect of the moon on the moon would be neglible on much longer timescales.
