--Resolved--
Another weird one. It's stuck on wanting Akeeba enabled, which is a little concerning, but deleting and uploading the latest com_joomlaupdate allowed it to work.
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This is an interesting error. Attempting to update a site from 3.8.12 to 3.9.1. Akeeba Backup on Update is active. After backup, I get a generic AJAX error. If I completely disable (not uninstall) Akeeba Backup, I get a component 404 when I attempt to update Joomla. I've noted that the URL still tries to use com_akeeba. Any ideas on what broke?
--update 1--
Clearing both site and browser cache shook out Akeeba and now I'm back to the AJAX error.
--update 2--
Joomla also can't self-identify at Control Panel under Maintenance. "Unknown Jooma, Unknown extensions".