Any attachment that isn’t fairly close to the plane of orbit is going to be bad. An entire hoop perpendicular to the main ring had better not be attached. It needs to rotate on it’s own in an entirely different direction to stay up.
Masscons and gravity from the Sun and Earth will cause more than up-down perturbations.
Did you watch the video by Isaac Arthur, or read the description on Wikipedia? The dynamic orbital ring concept allows you to compensate for the effects of perturbation, by adjusting the strength of the magnetic fields deflecting the mass particles. In theory, anyway.
If Earth’s tidal forces are a problem, couldn’t you position the ring so everywhere on the ring is equidistant from Earth at all times? Since the moon is tidally locked, shouldn’t that part be easy? Earthlings would see the entire ring circumscribe the moon at all times.
An unattached ring will circle in the same plane. Since that plane is not rotating wrt Earth, then it will appear to turn around the Moon once a (lunar) month. It’s little different from a Foucault pendulum but keeping in mind that a day on the Moon lasts a month.