The ticket is by AA. And I have the BA record locator too. BA told me again that the codeshare is the reason I can't check-in online (as I said, I have checked in online similar codeshares dozens of times with no issues).
Anyway, neither BA nor AA have found anything wrong with the reservation or the ticket. At least BA managed to assign me a good seat, so that I don't get stuck with leftovers by the time I show up at the airport.