Sorry for my potential idiocy. I think it came down to the fact that I left WC Cache running while I was troubleshooting all the other plugins. Once I left it off, I was eventually able to track down the culprit: an old plugin called “Frontpage category filter” (which wasn’t even doing anything so far as I could tell).
Everything seems to be fine now, no problems with Custom Community or the cache.
Yep, I ran all the repair options per Buddypress’s instructions. I ran them a second time after I discovered this issue. I created that test forum after that point, so I ran them again just now for fun. Still no change. 🙁
The only next steps I can think of are to hit that ominous “Reset bbPress” button or dig up my backups and just try again.