If I were you I’d probably just refund it as soon as possible, if it’s still possible - to save myself from having to deal in EA and their launcher in any way, shape, or form.
It’s already difficult on Windows and with Steam Deck/Proton it’s only gonna be even harder to figure it out, assuming if it’s even possible.
But a lot of the time it just won’t work, especially when a game has anti-cheat.
You can maybe try SyncThing but if the problem lies in your local network, this won’t help you. You’ve already listed your best option.