I've done something similar to this, however I don't see how mod organizer would work. The thing mod organizer does is use a "Virtual Install" basically, it never directly modifies the files of the game aside from putting the esps in the data folder. That means you'll need mod organizer in order to launch the game modded. But since mod organizer would be in a vm, that wouldn't work. In my experience, there is 2 ways of modding skyrim
1. Use a mod manager in wine (usually unstable)
2. Use Nexus Mod Manager LEGACY (its important to use legacy as its the version without virtual install) in a virtual machine with shared folders. The only issue with this is load order, but you can simply copy and paste the load order file (found somewhere in %appdata% in the vm) and copy it to <wineprefix>/users/<if you use proton, open steamuser, otherwise open the folder with your name>/Local Settings/Application Data/Skyrim/ and then launch SKSE.

The method I personally use right now is I run Nexus Mod Manager Legacy with Wine, which is as easy as using winetricks and install dotnet452, and then installing nexus mod manager legacy. Its pretty stable, though I found it crashes when it see's the script extender. Though you can simply rename the script extender executable while your doing stuff in the mod manager, and then when done, rename it back to what it was.