Argh, it took me a while but I kept on searching the archive with different words for that center brake light - came upon this:
http://www.peachparts.com/shopforum/tech-help/261365-no-power-3rd-brake-light-w210-1997-e320-2.html
Indeed, I didn't remember that the whole assembly unclips - one of the springy contacts on the base had corroded. I was able to bend up enough or the remaining metal tab to re-establish contact with the mount, but it looks like I might need to fashion something more permanent or find a good replacement unit.
I'm still on the hook for the left taillight socket bracket, and now know the third brake light is not on the bulb check diagnostic.