This is a really tough situation on a 10 hour or so flight. I don't know that you can safely knock a cat out for that long. In any event, I would relegate a carry-on solely to the task--clean towels or saint-smelling T-shirts as bedding, oversized zip-lock bags to put them in when soiled, unscented baby wipes (not kidding), a small amount of favored snacks, and a 5-10 milliliter dropper with which to administer water. It might be worthwhile to ask the vet whether one of the powdered milk-replacement formulas used for kittens would be an idea in terms of providing dense nutrition while avoiding nauseation.

There's a few paragraphs about sedation at the ACFA site that may prove a starting point in this regard. Best of luck.