After a few reports of plugins not updating correctly, it was clear a solution for updating mod files was needed that existed outside of the BepInEx plugin. Initially, I had thought a separate binary would be necessary, but boy was I wrong.
Introducing Preloader Patchers: These are loaded before plugins and are intended to allow mods to patch game methods before they are loaded. Instead I'm abusing the fact that they run before plugin load to replace all of the files from a previously downloaded update. Essentially the install step now happens after the next start of the game, instead of just before exiting.
This means a new file has been added to the client mod at BepInEx/patchers/Corter-ModSync-Patcher.dll. Make sure that file's there and you should be good to go!
Please work, please work, please work, please work...