you dont have to be logged out.
i'm lifetime plat on AA but credit all flights to BA now for a few years so when i need to fly AA, i use the AA app and am always logged in to my AAdvantage account and use the app to get my AA mobile boarding pass and yes the app is pretty fantastic on day of travel. that said, because my AA flights contain my BAEC number, sometimes it will just disappear from the app, so i have taken to ensuring i include the AA PNR somewhere for my easy reference when i need to look it up on the app. i am ALWAYS mindful to ensure that my BAEC number is on all AA flights since it can be a mess afterwards to credit if a larger amount (not fussed over 40tp, am fussed over 140).
and i use the BA app for any BA flights.
...and us the BA website online to track that my AA flights are appropriately credited since i find the BA app generally lacking.