We are getting closer. Now the game actually starts via Proton. However, the entire control system seems to be blocked. Both the directional pad and the joystick do not respond to anything.
So... A bit of an update on that side (sort of). I've noticed that the Steam Overlay also didn't work after packing, as the required command line argument wasn't passed to the game's executable. Although I'm not sure if these are correlated, but it's possible that there could be a conflict.
I am working on adding the required flag automatically for the Steam builds, but until then, you can fix that (if needed) by adding the --in-process-gpu in the executable flags (found under Project Settings -> Compiler and Packer -> Halva Settings).