Actually on Anerican, you CAN NOT!!! Unless they changed the rules in the last year. On INTERNATIONAL flights to BA the only airlines that I know for sure allow this (with advance reservation only two pets in the whole cabin and only in economy) are DELTA (which is VERY pet friendly...even have a special check in counter for people traveling with pets in Atlanta, their hub) and UNITED. American only allows pets in the cabin with a note signed by an MD specifically stating the animal is "an emotional assistance pet"...no ifs and or buts. I showed up at Ezieza with a paper that said she was a therapy dog and was not allowed to board. PLEASE someone correct me if they have changed their policy. I would be very happy to hear that! Remember domestic and international rules can be different...
That aside, when returning to BA with my little dog and three suitcases everyone was VERY helpful and wheeled my cart right past customs with the doggie atop after stopping at SENASA.