I doubt anyone will actually need this guide during my lifetime, I'm putting it here as a reminder to myself in case I ever have to go through this process again.

1. In Desktop mode use a browser to download the Ubisoft Connect installer.
2. Open Steam (Desktop Mode) and in the bottom left click "Add non-steam game" and navigate to the Connect installer in your downloads folder.
3. Right click on the installer entry in Steam and go to properties/compatibility.
4. Force compatibility with Proton 7 or newer.
5. Run the installer through Steam.
6. Add the Connect exe to Steam, set its compatibility as above, and then run it. It is located in home/deck/.local/share/steam/steamapps/compatdata/{big number}/pfx/drive_c/program files (x86)/ubisoft/ubisoft game launcher
7. Install the Uplay version of TFS, the Steam version does not function.
8. Launch the game to make sure it works.
9. Switch over to Game Mode and launch Connect, then use the mouse controls to launch TFS.
10. Press the Steam button and edit the controller settings to achieve left-click for menu speed, etc.