Thanks to this community and comments from fireaphid, I was encouraged to try beveling the edges so the stem would resemble this: \_/ more than this: |_|
My husband helped me with the beveling. We used both a metal nail file and an emery board. We found that the stick of a Q-Tip cotton swab (with all the cotton pulled off) was the perfect size to slide into the Moon Cup's hollow stem. This kept it firm and steady while we filed. We used a nail clipper to clip the Q-Tip stick off at the end of the cup's stem, which made the filing easier.
It worked so beautifully that this cycle (my first with the beveled-edge stem), I cannot feel it at all. I keep getting the urge to check and make sure it's still there.
I really like this community, and I hope this story helps someone else with her own stem!