[Sound of dialing American Airlines reservation agent on my cell phone]: "Blip - blip - blip - blip - blip - blip - blip-blip-blip-blip."
"Hello, can you tell me if you fly dogs to Panama City during the evening when it's below 85 degrees?'
"Oh yes, we do that all the time."
"Good. The rules on-line don't make that distinction. I see you have an evening flight from Miami to Tocumen that leaves at 6:00 pm and arrives at 8:30. But I don't see how to get from BWI to Miami in time to catch that flight the same day. Oh wait, what about our other airports - Reagan and Dulles?"
"We have a flight leaving Reagan that gets into Miami at 4:00 pm."
"That's perfect - book it for Friday please."
Cost of taking the dogs this way: $300.
Done. In the end, it was so simple. It just took me about a week to figure it out. And they even have Wi-Fi on board:
Muchas gracias a mi amiga Kris para la información sobre los vuelos por la noche.