I don't have experience with that game specifically, but generally most games are easier to run through steam than through a 3rd party store. You often have to manually identify missing .dll files and install them through winetricks/protontricks to get games to work. The EA launcher in general doesn't play nice with Steam Deck, so I don't know if you'll be able to get it working.
It's worth noting that the steam version of Battlefront 2 didn't work well either for a long time, due to the EA app. My understanding is it works now, but the game still has some general issues with crashes and stuff. Some recommendations people made are to run the game in windowed or borderless (not fullscreen), to let the game sit on the main menu for 30 seconds before trying to play anything, and to maybe try forcing proton 9 instead of proton 10 or experimental.