I always fly American with Pico because my daughter has flight perks from her job that let me fly almost free. However, I have read in the Wall Street Journal some time ago that Continental had the best pet cargo care and it surely seems as though that extends to the in cabin pet privileges, also.
I pay $40 for each leg of the flight on American. So my flight to Florida, round trip, costs $160 to take Pico AS CARRY ON LUGGAGE! I get absolutely no value for that as he has to stay in his carrier and on the floor under the seat in front of me. It is just a way for the airline to increase revenue at no cost to them.
Plus another $44 to get his health certificate (a new one each time he flies) and it runs $200 to take him with me.
That said, I have a few tips for you that work even when flying on American Airlines:
Choose your wardrobe carefully to avoid having to undress. My travel shoes have no metal in them and my travel clothes have no zippers or metal buttons or anything like that. No belt, either. This allows me to zip through security with no hassles.
Take your furbaby out of the carrier (I don't put his harness on until we are through security) and put the carrier through the scanner, holding him in your arms to walk through the detector. At this point you will be mobbed by the security people wanting to know what kind of breed you have and telling you about their darlings, and oohing and aahhing over your baby.
At the gate I put Pico on harness and leash in a remote uncrowded area so he can move around and sniff and generally be active until boarding.
When boarding, I hold his carrier high so the flight attendants can "see" who is inside. If they smile and go Ooooooh, I'm pretty sure I can take him out of the carrier for at least part of the flight. If there is no response, I'm going to have to be a bit sneaky if I want to take him out. Because I am flying Standby, Non-Rev, I have to adhere strictly and without arguing, to whatever the attendants tell me.
Funny story, on one flight I was bent over in my seat talking to Pico to reassure him until takeoff. At some point my seatmate, a young man, asked if that was my child in the seat in front of me and if I wanted him to switch seats so we could sit together. He was amazed to find I was not talking to the child in the seat in frontn of my but to my sweetie dog in the carrier on the floor!
I always ask my seatmate if they would object to my taking Pico out. I've never been refused. He usually settles down pretty quickly after getting acquainted with everyone within reach and curls up in my lap and sleeps the rest of the trip. Most times the flight attendants don't even notice him, thinking perhaps that he is a sweater on my lap or something. Other times I will partially cover him with a blanket.
You must have him in his carrier, on the floor under the seat in front of you for takeoff and landing.
On connecting flights I take him out for a pee break. It only takes about 20 minutes to go out and then back in through security again at DFW. I have found a little patch of grass on the Arrival deck where he will go. Otherwise, he won't go in the jasmine ground cover.
I use Cheerios to keep him occupied if he is very unruly, which is seldom.
I love flying with Pico and hope you and Bella have a great trip, too!