Could not login with EA launcher using the Steam deck at all. We ended up creating an EA account on a mobile, then linked it to the Steam account we were playing on.

After doing this launching the game breezed through the launcher without issue.

Not sure if changing proton version was relevant, can't go back to try.

During launch the secondary EA launcher was complaining about syncing issues, options to resolve did not inspire confidence, shutting down the game and trying again launched the game without issue.

The side of the splitscreen you play on is determined by the character you choose, not the order of controller 'signup'