DARQ has just been updated to version 1.1 and Nintendo Switch Pro Controller is working now.
The games changelog mentions:
In addition to Steam Controller, DARQ now supports X-Box One and X-Box 360 controllers, both wired and wireless.
Appears this also fixed Nintendo Switch Pro controller at least when using Proton. Closing this issue, as I'm no longer seeing any problems (and the controller not working appears to have been an issue with the game itself).
proton 4.11-2x1 2019-08
Compatibility Report
System Information
I confirm:
Symptoms
Game works with good performance:

This is "fantastic" quality setting at 2.5K resolution, about 75-100fps depending on scene
The only problem I encountered is that controller input is detected. This was when using Nintendo Switch Pro controller over bluetooth.
This can be an issue with the game itself and not proton, as there are reports on the Steam forums from Windows users having controller issues.
steam-433550-proton-4.11-2.log
system_info.txt
Reproduction
Just launch the game and try to use the controller