I wonder if you could map the writes to the drive to some sort of permanent storage programatically.

You could. You’d just need some kind of file system driver that connects the application to a more stateful disk so as things pass through temporary memory they’re written to an SSD or HDD.

Of course that already exists, and DO offers it for $5.

This can be done using overlayfs, but it kind of defeats the purpose since you now need a storage device