1. Install Fallout Fixt normally into the Steam Fallout directory. After you are done you should have a Fallout Fixt folder in the Fallout Steam folder.

2. Delete all files that Steam installed for Fallout, EXCEPT CRITTER.DAT, MASTER.DAT, Fallout Fixt folder and the DATA folder from the original Fallout Steam version (NOT INSIDE THE FIXT FOLDER!). Those files Fallout Fixt needs to launch and run everything properly.

3. Both CRITTER.DAT, MASTER.DAT and the DATA folder have to move one directory lower. Here is the path it needs to go to: "C:\Program Files (x86)\Steam\steamapps\common"
Don't be wondered why those files should be there. It's just that the FalloutW Fixt exe loads them from one directory beneath. Its a bit odd, but that is how the mod is designed.

4. Move all the files from the Fallout Fixt Folder to the Fallout Steam folder directory. You can delete the Fallout Fixt folder inside the Fallout Steam folder after this.

5. Rename FALLOUTW.exe to FalloutLauncher.exe. Because this is what Steam launches by default. If you don't do this you can't launch the game over Steam anymore.

6. If everything has been done properly you should now boot up Fallout Fixt when starting the game over Steam!

Now you can enjoy Fallout Fixt with Steam tracking your time.

This is honestly an unusual approach but it works. Was in fact the only way I could get it working.
I hope the Fallout Fixt developer makes it a bit better for the Steam version in the future. Wouldn't even mind if the mod overrides all Fallout files. This would make it easier for people to launch Fallout Fixt with Steam.