So ... I fixed it! Or at least found a workaround. I wanted to update this for anyone who has the same problem in the future.
(To answer your question, I'd left mods and CC out of my game until I had absolutely given up on fixing it.)
The way I eventually solved the problem was by creating a new Windows profile and running the game from there. The game now takes screenshots again, even with my preexisting save. I don't know why it doesn't work on my regular profile (even after a second complete uninstall and reinstall, it still doesn't), but it doesn't matter anymore. I now have a separate profile just for TS4. I think I saw this suggested as a way to fix errors on some EA page, so kudos to them! I wish I'd tried it sooner.
Thank you for your help, Sasha!
