You have to instal dotnet472

$ protontricks 359320 -q dotnet472 win7

This takes about 30 minutes with a lot of error messages. After it finishes you hav to make modifications to config files in the game directory, from both:

/steamapps/common/Elite Dangerous/EDLauncher.exe.config /steamapps/common/Elite Dangerous/HardwareReporter.exe.config

Delete the following line: <supportedRuntime version="v2.0.50727"/>

and add the following: <supportedRuntime version="v4.0"/>

After changing resolution, change the option to fullscreen mode. After a few tries it should work.