Many thanks for all the contributions and ideas.
A couple of notes: shooting from above I assume would be difficult, requiring a copy stand and a completely different lighting arrangement. It would also create different shadows and reflections around the ring, creating an impression that the ring is standing against a wall. Since most rings require two or three views, one vertical, one horizontal, and perhaps one at a 45 degree angle, if you are shooting from above, that would again raise the issue how to position it for other views. Changing from "above" position to a regular wold also be impractical when doing hundreds of pieces.
Fishing line might work, but tying and untying it each time would be cumbersome and time consuming, I presume.
In the meantime, I acquired a crystal clear museum wax and am getting better at applying ever smaller amounts of it: