Online Check-in will probably not work, even if you use VeriFLY.

AA will verify the PLF and other documentation. They will likely print boarding passes for the entire journey if they know what they’re doing. If they process things correctly, BA won’t do a separate document check but may ask to see your Covid test or PLF upon boarding. If the AA agent is not up to speed, your name will get called by BA.

If I recall correctly, because LAX has biometric boarding with BA they may insist on giving you a new boarding pass, whereas if you connect somewhere like DFW the AA printed pass will let you board.

I’ve had 3 similar journeys during Covid from PHX and Check-in took maybe an extra 2-3 minutes.