I do not understand the problem.
You need an EA Account to Play the game. Your EA Account is linked to Steam for that purpose only.
You bought the game on steam store, your are starting the game on steam therefore the game is using your steam profile.
https://tracker.gg/bf6
The Tracker has EA, Steam, Playstation and XBOX.
Your Ingame Progress should update Independend of the Store / Launcher because its linked to your EA Account.
Everything is getting logged into the ea account no matter which platform you use.
