The Last Resort: Wipe and Reinstall
Unfortunately, the fix I had only works part of the time, but the fans were still running pretty constantly. This was bugging me more and more, so I ended up biting the bullet and opted to wipe my machine, re-install Catalina and work from a time machine backup.
A little daunting at first, but I used this guide and was able to step through it fairly easily. The actual install of Catalina wasn’t too bad, honestly. But the updating off of Time Machine was what took several hours.
The fans are still not… great. But they’re better than they were. At least now it’s only kicking on when there’s intentional, intensive use… and not just all the time in the background.