I had a cat that got jittery in the car too. I drove with him from California to Minneapolis. The vet gave me sedatives and this helped calm him down. He meowed a bit but at least he wasn't scratching around a lot and trying to get out of the cage.
I travel by plane a lot and have taken my kids on planes without my husband. If it was me I wouldn't take the cat on the plane. I would beg my husband take my cat with him. He'll have more flexibility to let the cat stop for potty breaks, etc and generally the sensation of flying is worse than that of riding in a car. The logistics of checking into the airport, going through security, etc. are tough just with the kids. Adding the cat will make it go much slower, making it more likely you and your daughter will get irritable too. Besides when you consider adding the time to drive to the airport, check in, fly and pickup luggage it's not that short of a time for the cat to be stuck in a carrier anyway.
I can't imagine being stuck on a plane for a few hours with a fussing child and airsick, screeching cat.
For the next three weeks, I'd take the cat on car rides to get him used to riding in his carrier and the sounds and vibrations of the car. If he gets used to the sensations then maybe he'll be less upset. One of my friends takes her cats to her mom's when she travels - the first time they got irritable now they low going on car trips. For your best advice ask your vet!
Good luck...