Don't forget that if you go through LHR, you and the pet will have to remain airside due to the UK's very strict quarantine and rabies rules, unless they are microchipped, and have a pet passport. More information on the UK rules are here.

Although going through LHR might be quite a good idea, as we have a TV programme over here called Airport which is one of these documentaries following round people at Heathrow. For a couple of seasons they followed round the people at the "Animal Reception Centre" where they deal with all animals in transit. They seem to be very knowledgeable, caring and extremely well equipped for dealing with everything that comes through the airport. Ok you'd expect them to be, especially if they're appearing on a TV programme, but I was still impressed. They must have appeared on at least ten of the programmes in the series now, although I don't think they're making any more now.

Edited to add that another good reason for going through the UK is that we never have any good weather during the summer, so it's pretty likely that it's going to be grey, miserable and less than 85F!