Saw that episode of Airline, actually. SWA's pet policy? They don't have one since they don't allow pets. Period. Almost every airline I know of charges a fee for the pet to ride in-cabin, so there's really no getting around it. You can call around to the res centers and ask to get the policy of each, and that's the only way to be sure. I was about to mention Companion Air, but someone beat me to it.
Nole, why the downer on them? They're a legit 135 op out of Boca that specializes in animal lovers, and they make a killing off of people flying to dog shows. They have kennels built into the plane, so it's a lot more comfortable for the animals than a cargo hold or being stuffed under a seat would be. Their mins are the standard 135 1200 TT, so it's not like they're a shady time building op. It's like saying that all Flight Express needs is a time building program.