Thanks, cepheid and obscure2k and wharvey.
I'm pretty well decided that I'll take the pets with me on a non-stop redeye, even if it means flying to IAD or DCA or ATL rather than CLT. At least the animals are under my direct control in a rental car which is air-conditioned, for the drive to our destination in western North Carolina. And I will make sure the elderly cat and the part-Persian are with me in the cabin, maybe taking advantage of Alaska's offer of carrying on two pets if I buy two tickets. That way, each could have his own carrier.
CLT is the closest airport, but US states no pets in the hold during summer. I think I will double-check, as a non-stop redeye won't involve any extreme temperatures.
I think I will likely do this in two trips by air. I just did the cross-country drive (no pets with me on that trip), and learned that it is not easy to find a hotel that will allow more than one pet per hotel room (and they often charge extra "cleaning fees"). The cats hate being in a carrier at all, so I am figuring that 5 - 6 hours on a plane (even with a car trip from the airport to the house) is better than 5 - 6 days on the road. That was also really pushing it for me, at 600 miles per day, let alone for pets in carriers.
My main concern is the safety of the pets, cost is only a distant second.