Well I started thinking about it more.

If it's orbiting at the speed of the moon, it's going slower than geostationary orbit speed. So its tendency will be to fall towards the earth. But since it's attached to the moon, this won't happen.

Not sure how much station keeping would be needed to hold this in place.

Also will have to deal with collision risks with objects in geostationary orbit. Though I don't think geostationary orbit is a particularly important factor. Could probably be a few thousand Km above geostationary orbit without any significant adjustment to their plan.