You would need to replace a few things but the UX you would just build in to the game's update process.
The UX would be something like:
User starts game
gets notified of update
clicks to download update
update downloads
update prompts you to install (with notification that game will close and restart)
and on that click the installer would run, it would do whatever you hade made it do, the game would close after the call to run the installer, then installer would run the game again after it finished updating the files.
I don't know of many programs that can update themselves without closing and restarting, it's not an issue.
EDIT: Completely agree that Steam's Dev UX is trash for small developers