For anyone running into this in the future.
I got the game running using this guide: How to install and run Cod4x on Linux (Steam)
Then the config file is here: ~/.local/share/Steam/steamapps/compatdata/7940/pfx/drive_c/users/steamuser/AppData/Local/CallofDuty4MW/players/profiles/[yourProfileName]/config_mp.cfg
and using the original config file I can open the ingame console with:
Alt + AltGr + Circumflex(^)
(on a German keyboard layout). I need to press all three. No idea why - some wine or proton stuff maybe.