The bag remains with the original (0)001 number, who prints the boarding pass is irrelevant; as long as its itinerary is printed it’s not an issue.

Having said that, BA and AA do continue and will continue to have issues in “passing” the baggage info to one another. It’s more an issue for BA at large AA hubs, for American’s Sabre DCS system is a lot less good as Altéa in sending out forward messages. But that has nothing to do with which boarding pass is used.