I tried a lot of things that didn't work. Flagging, defending myself, patiently explaining my point of view ad nauseum, request that the other moderators in the same SE site take a look at the contentious exchanges, bringing up problems in Meta, going to Chat, rolling back to my own answer text when I felt over-edited, showing my respect and appreciation for certain aspects of the person's work, requesting that the site moderators bump the disagreement up to a higher power. None of those things helped in the slightest.
Here's what did help: I hit the "contact us" link at the bottom of the page, and laid out my point of view in a respectful tone, but with honesty.
Apparently, what this accomplished was to get a problem escalated -- which I had not succeeded in accomplishing by any other means.
The first response from the SE Team showed pretty clearly that the team member hadn't looked at the situation very closely. But the system allowed me to write back. I provided a couple of specific links that I thought illustrated my points clearly.
Ready for the happy ending? The SE Team member, as a neutral third party, was able to look at the situation objectively, recognize my good intentions and my willingness to continue to learn and improve my effectiveness as a SE participant, and have a respectfully critical talk with the moderator I was in so much conflict with.
Sometimes two people push each other's buttons. It doesn't mean either of them is a bad person, or that they'll never be able to get along. But it can be helpful for someone completely unconnected with either party to take an objective look at the situation.
I am writing this not with the view of further embarrassing myself, and certainly not with the goal of embarrassing anyone else, but to document an approach that works.
I only wish I had figured out this approach earlier, before things got so polarized.
This site has a lot to offer, and hopefully there is room for lots of different parenting styles, writing styles, and human being styles.