SteamTinkerLaunch worked like a charm! Here’s what I had to do to get it to work, for anyone who stumbles across this post:
Install the package pacman -S steamtinkerlaunch
Run the command steamtinkerlaunch compat add to initialise it
Add it as the required compatibility layer in the game I wanted (SH3) then restart steam
Launch the game - the steamtinkerlaunch window opens. Click on the MAIN MENU button before it launches the game
From the new steamtinkerlaunch window that opens, click on “One time run”. This opens yet another window
In the field “One time command” navigate to where I had LSH3_2022 downloaded, and select it
Click on RUN COMMAND
Follow the install steps in the LSH3 manual
Repeat the steps related to “One time run” again but this time select the JSGME installer, and go through that installation
Launch JSGME again using “One time run” and enable any mods I need (although LSH3 has a 1920x1080 widescreen mod included, this doesnt seem to work at least on my setup. I had to enable the Ahnenerbe WideGui 1920 x 1080 Final mod)
Launch the game. It should work.
So far I completed the first training mission to make sure most things work. Hopefully there aren’t any issues further on down the road. Thanks for the suggestion, I had no idea this tool existed, but now I will probably be using it quite a bit!