Hello there! I just managed to install and play Falcon BMS on my Linux machine (Arch), and as such I'll make a brief tutorial on how I did it.
Step 1: Buy Falcon 4.0 on Steam
Step 2: Open Falcon 4.0 on Steam
This is so the BMS installer can recognize that you aren't pirating Falcon 4.0.
Step 3: Close Falcon 4.0 after it reaches main menu.
Step 4: Download latest Proton-GE and install it on your Steam install.
This is a custom Proton fork by Glorious Eggroll. It is better than regular Proton for some reason.
Step 5: Run Falcon BMS 4.37 installer through Protontricks in Falcon 4.0's prefix.
If you haven't installed Protontricks, install it, since it is necessary for this tutorial.
Step 6: Update Falcon BMS to 4.37.2 using the updater through Protontricks (Falcon 4.0's prefix).
Step 7: Play Falcon BMS using the regular launcher (The alternative launcher did not work for me) through Protontricks, using Falcon 4.0's prefix.
Now it is done and you can play Falcon BMS.
Now I just need to figure out how to use TacView and Simple Radio Standalone so I can finish this tutorial.
edit: Tacview is pretty simple. Just install it using Protontricks and launch it also using Protontricks (on Falcon 4.0's prefix, as always).