Theoretically you could use LIMO:
https://github.com/limo-app/limo
I have no idea how to use it yet.
It can be done with Winboat. I just finished my first experiment and tried Skyrim Special Edition with the Unofficial Patch and SkyUI. It worked, but I had to set up all the file paths manually since the game isn't installed on the virtual machine. Winboat lets you access your Linux home directory on the Desktop so long as you enable that option while setting up Winboat.
That and at least with Skyrim, you need to run the game once on Linux first to generate the initial INI config files. Since it's not installed on Windows you need to copy them out of the Steam prefix folder and then paste them where they're expected to be on Windows. In Documents/My Games.
Once done modding, copy those INI files if any mods touched them and overwrite the INI files on Linux.
In Steam, set Skyrim's launch arguments to:
bash -c 'exec "${@/SkyrimLauncher.exe/skse_loader.exe}"' -- %command%
Then you'll have Steam Overlay with Script Extender loaded.
Messy screenshot gallery: https://postimg.cc/gallery/6WGPD4h
Sorry bed time here and I'm learning how to do this on the fly. I didn't organize this very much.
Given this success I expect Mod Organizer 2 and METweaks Mod Manager for Mass Effect would work too with some annoying babysitting.
edit: One thing that really annoying is Steam using the Title ID numbers for compatdata instead of the fucking game NAME. It's not human friendly to navigate. Have to look up the TitleID on steamdb or sort compatdata folder by Modified most recently in Dolphin or whatever.