I do not feel totally convinced that the carseat is necessary for the plane safety--I'm willing to take my chances. But if turns out that 9 hours with the carseat is miserable, I can gate check it for the other flights.
Do NOT check a car seat as luggage. They get damaged and lost by the airlines all the time and compensation is rarely offered. If it gets lost, you wont have it. You'll have your baby in your lap in the car. End of story.
You may not feel "totally convinced" but I do. I've seen all the scary crash videos as a Flight Attendant! Your feeling contradict facts. A car seat can make the difference in an accident, especially on take-off and landing when most happen. The dynamics are similar to a car, except the plane is going faster!
You don't have to keep her in the car seat the whole time. Some parents believe this but it's not true. Mine were in the car seat for take-off, landing and when they slept. Otherwise they were in my arms (often nursing!) or we were doing "aisle laps" (I had very active toddlers!) It was nice to place a sleeping baby down and get some sleep myself on an 11 1/2 hour flight... When the seat belt sign came on, I didn't have to do anything and knew that my baby was secure.
Another of my tricks was to breastfeed during boarding, right before putting them in their car seats for take off (for the record, the plural refers to multiple children, didn't manage to tandem nurse). Often they fell asleep rolling out to the runway and were fast asleep before we even left the ground. This was often the case at big airports where the taxi can be long. Is she then wakes up after take-off, you can remove her to nurse. You're supposed to keep them in the seat until the seat belt sign goes off but once you're in the air, there's much less risk.
Someone mentioned the double seat belt. These are banned on U.S. companies, by the FAA, for safety reasons. Just like in a car, having the baby attached to you is dangerous because you weigh more than the child and in forward impact, you would come down on them.
I used a foldable metal luggage cart and attached the car seat. I could then pull this behind me. The advantage was that my kids weren't big on strollers so bringing one didn't make a lot of sense, especially once they walked. They liked walking but then would get tired. The car seat could then be used to pull them through the airport when they got all floopy and cranky, but only when they just wouldn't walk or didn't want to be worn. Between a good baby carrier and this car seat contraption, it's easy to get through the airport, even alone with three kids like I was on many, many flights!