Install Lutris's flatpack via Discover
Use this install script https://lutris.net/games/install/30919/view to install the wine prefix with needed dependencies
Open the recently installed Wine's configuration, and set all of the xinput library overrides to builtin, native.
Add Lutris or the recently installed Modern Warfare 3 to Steam as a non steam game
Play in game-mode, steam deck controls should work (hold Steam button and use touch pad to move and click mouse)
For MW3, for calling in killstreaks, one could add a radial menu to the left touch pad to press 4, 5 and 6.
Beware of installing other apps in the wine prefix (like steam), it could break xinput for some reason.