I contacted Jatol, my host, and asked their advice. It was too late to recover the files from their nightly backup, but they suggested I try the “add a new user to the database” route. So I added a new user, gave it permissions, then altered wp-config to match. I ran upgrade.php, and got a success this time.
Tested the blog out, changed the theme, and everything seems to be working okay. I wonder if I maybe was using the wrong password before.
Either way, whether I used the wrong password or the one-click upgrade messed up, it’s working now, and that’s what matters. Just thought I’d post with my results in case it might help other people.