This might be the same bug that I reported in ValveSoftware/steam-for-linux#6436, but it is unclear from your report whether it is the same.
There is a similar bug affecting ARMA 2, but the registry keys get reset on every start, so unless you launch the game directly, you cannot play it. :/
Nothing extracted yet.
Compatibility Report
System Information
I confirm:
steam-65730.log
Symptoms
Game fails to launch - product code not added to Windows registry. Splash screen loads then game crashes to desktop with Bad Serial Number error
Reproduction
Install Take On Helicopters on Linux, then launch.
I have managed to fix the issue manually by installing Take On Helicopters on Windows 10, then running regedit and exporting the key HKEY_LOCAL_MACHINE\SOFTWARE\WOW6432Node\bohemia interactive studio.
I then imported this key into the Linux version and launched the game successfully.