The same thing occurred to me at LGA 12-18 months ago when I jumped from Bronze to Silver post booking on AA, and AA didn't have the updated status. The BA app card was not enough, and I was refused entry. I didn't push it.

The irony is that when you scan your AA BP at the Flagship Lounges, they ALWAYS ask me verbally "What status are you? and who with? BA?" - so they rely on the scan and then a verbal follow-up and some kind of keystroke (in which case why can't you accept the BA app status/card as a backup?)