As bisonrav has stated, there are a lot of considerations when an airport has a short runway. Florence’s is only marginally longer that LCY. Add in a heavy aircraft and high temperatures and (quite rightly) the pilot will have to make changes to reduce the weight of the a/c. Now a separate consideration is how well it was handled by BA. Sounds like not very well. I would have thought that, given the short runway, they would have established a plan for these sort of events. However, since BA clearly knows the amount of passengers (and luggage) booked on the flight, it should not have happened in the first place.