Got it open! The outer and inner plastic discs were quite firmly stuck together, but I was able to get them apart after running a scalpel blade in between them, around the whole circle. Put on some antistatic gel, let it dry, and it does seem to be improved. Not certain it's quite right, still, but much better.
The ring with the membrane was stuck to the inner shell as well, so I only coated the top side - maybe that's my problem. Didn't see a clear way to pry it loose without risking puncturing the membrane.
Thanks much for your help, spritzer!