Only one location for it to go: smack in the middle of the "dark side".

It made intuitive sense when I wrote it, but it's an odd thing to think about. If you attach the thing from the "dark side" it seemed to me that you include a little bonus velocity from being on the extreme outside of the lunar orbit around the earth. But on the other hand, if you attach it towards the earth, it has exactly the same velocity around the moon, and exactly the same distance from the moon. So the moment should be the same.

Additionally, if you attach it towards earth I think you would get a bonus from earth's tidal attraction? But knowing what we do about tides on earth, the tidal effect should be about as great on the opposite side of the moon, too (because the earth would be pulling the moon away from the 'vator counterweight)?

My head hurts!