Arma: Cold War Assault (bad serial number in setup) (65790)

Your system information

• Steam client version (build number or date): Aug 23 2018, at 19:47:27
• Distribution (e.g. Ubuntu): Manjaro with Xfce
• Opted into Steam client beta?: [Yes/No] Yes
• Have you checked for system updates?: [Yes/No] Yes

Please describe your issue in as much detail as possible:

Arma: Cold War Assault doesn't detect the game's CD-key. Steam starts the game, I get a split second of the Bohemia Interactive logo to show up and lastly a popup windows is shown with "Bad serial number is given in Setup".

Steps for reproducing this issue:

1. Just start the game in Steam with Steam Play in Linux.

More information:

I tried validating game files, but the error remains.

After some research it seems the game stores the CD-key in the registry:
HKEY_LOCAL_MACHINE\SOFTWARE\Codemasters, but there is no way that I know to enter the key somewhere.

Start by reproducing the failure by launching Arma: Cold War Assault through Steam Play on Linux, then inspect how the game's CD-key is handled under HKEY_LOCAL_MACHINE\SOFTWARE\Codemasters. Use the reported bad-serial-number popup and the failed file validation as the baseline; done means the game detects its key and launches without that error.