I assume this would only work for mods that don't touch the story editor, correct
I haven't tested that, but if it could work, you would put your story files in
Mods\Shared\Story, like where they are in the Shared.pak. There may be some merging required depending on how the story scripts are loaded.
The small mod that I figured all of this out on required a lot of debugging to figure out what was causing issues, so I probably wouldn't make a "loose files" version if the mod was big in scope, and added new levels, quests, story stuff, etc. Just seems like a headache to get working.
It would probably be easier to figure out a way to patch out whatever's disabling the achievements, like they did for Skyrim.