Yeah, the ring was in the box with the clock, and no, it couldn't be for the larger instruments. It reminds me of rubber bands that have sat somewhere for a long time and "go stale" - no longer "stretchy" and (in the case of rubber bands) break easily when stressed.
This instrument "seal" is floppy loose, just by a bit, but loose nevertheless.
I'm wondering about maybe cutting a short length and perhaps using some adhesive to join the ends, say at the bottom ?