Well, that was a different outcome. I tried to update our Mastodon install and it worked! But it lost all user accounts along the way.
I think at this point I may have to look at doing a database backup manually, then restoring and migrating, or something like that.
Failing that, we may have to look at hard measures for asking folks to take their own data with them and start fresh (with someone paid to keep us up to date).