1. It is unlikely that the agent at JFK would be willing to check the bags to LHR only, as the ticket's final destination is BUD, correct? Back in the days when they were doing baggage tags by hand, this was possible but nowadays the computerized system dictates everything
Nowadays the computer says ‘No’ because BA have programmed it say ‘No’ : this is policy to reduce revenue loss by travellers dropping the last leg of their itinerary.
Booking confirmations warn of this :