Could the Moon naturally capture a large asteroid and keep it as a subsatellite? For how long? (Edited at James K's recommendation to stay on-topic for this SE.)

From my reading on this SE and from Googling I've gotten conflicting reports on the viability of subsatellites this close to the Sun, but a naive application of the Hill Sphere equation suggests that if it got captured at all, a large asteroid could theoretically orbit the moon in a stable orbit.

What I don't know is this:


For how long? Longer than a couple hundred thousand years, such that humans on Earth wouldn't have known anything else? Longer than a million years? Probably not longer than a billion, due to the tidal effects decaying the asteroid's orbit, but is that a bad assumption?
How eccentric would the asteroid's orbit be? Would it be unlikely for it to orbit with negligible eccentricity, whether on the ecliptic or not? If so, what kind of eccentricity/orbital inclination would be reasonable?
I'm going to discuss only our moon, to keep this on topic.

This sort of orbit could be stable on the sort of timescales that you want.

Objects in Low Lunar Orbit will likely suffer from the irregular gravity of the moon. There are "mass concentrations" as a result of ancient meteorite impacts. These "mascons" distort the orbits of objects in low orbit, and cause them to collide with the moon in a relatively short time. However at 24000km the gravity field is smoothed by your distance from the surface. You should not have problems with mascons at this distance, at least not over 100,000 years or so.

The Hill sphere of the moon extends to about 60000km, so your orbit is well within the Hill radius, and so can be stable. 

The moon spins slowly, so tidal effects would tend to cause your asteroid to lose altitude, and eventually hit the moon. The timescale of this would be longer than a few 100,000 years 

Getting your asteroid in orbit is very tricky. You would need to hypothesise double asteroids transferring momentum between each other and the moon, and one being lost while the other ends up in lunar orbit.