Concur with the posters above, but adding the fact that married segment could help OP in this case. It could be that MIA-BON, on a standalone basis, only has full fare J available, but if it is booked as a through LHR-MIA-BON trip the entire trip could be booked in discount business class.

Since it sounds like this is the outbound leg, any change will cause the ticket price to be calculated based on the fare at re-ticketing (with some caveat with the return segment keeping the same fare basis code, etc). In other words, OP's could monitor the AA website to see if the entire itinerary someday became available in Business, and if so, the prices shown will be the new ticket price.

Actually, for real questions, you should call AA. The chat function is good for basic questions and simple changes, but for international itineraries, it's usually better to call.