Yes that was the unacceptable attitude I complained about, a BA top tier frequently flier who buys a J ticket LHR-SEA-LAX-LHR, with 2 BA segments & lounge access, expects not to be turned away and fobbed off with excuses like you’re flying AA today.
but this seems to be the crux of it.
You were there as you were departing on an AA flight so your lounge visit was as an AA passenger not as a BA passenger.
BA’s lounge has space for only so many people. BA choose to prioritise those BA passengers on the BA flight hence why you (and any other AA passengers) are turned away.
I guess if you have limited capacity it’s seems sensible to prioritise BA passengers over AA ones? I wonder what passengers on the BA flight would say if they were denied access because the lounge allowed those on AA domestic flights instead?