OK, so I managed to find how to convert the CD-key into a Hex-Key to put in the registry as described by another comment below.

User Gator96100 posted on https://www.unknowncheats.me/forum/arma-2-a/99874-release-multithreaded-key-checker.html a program with source code that could convert a CD-key into Hex for the ARMA series.

I did not manage to make the program run under Wine, but it worked fine on a Windows session and gave me the right key (same as an installation gave via Windows Steam, confirmed this by pasting the key in my Ubuntu ARMA Cold War Assault registry and managed to start and play without any issues)

The program is written in C#, which I have no experience in. If I can, I'll try and convert the script into something useful that can be run in the terminal on a Linux system, but I am a poor programmer. I will post the relevant part of the C# code below so others may help. Sorry for the mess, ProtonDB won't accept new lines, if it helps, get the code from the link above. CODE STARTS HERE:

using System; using System.Linq; using System.Security.Cryptography; using System.Text; using System.Windows.Forms;

public class KeyConverter {
    private static readonly String szTemplate = "0123456789ABCDEFGHJKLMNPRSTVWXYZ";

    public String getHexKey(String arma2key) {
        var upperkey = arma2key.Trim().ToUpper().Replace('O', '0').Replace('I', '1').Replace("-", "");
        var bResult = new Byte[15];
        if (upperkey.Length != 24)
            throw new Exception("Invalid key length");

        for (var i = 0; i < 3; ++i) {
            UInt64 qwResult = 0;
            for (var j = 0; j < 8; ++j) {
                var cChar = upperkey[i * 8 + j];
                var szPos = szTemplate.IndexOf(cChar);
                qwResult |= (UInt64)szPos << (j * 5);
            }
            for (var j = 0; j < 5; ++j) {
                bResult[i * 5 + 5 - 1 - j] = (Byte)(qwResult & 0xFF);
                qwResult >>= 8;
            }
        }

        return BitConverter.ToString(bResult); ;
    }