As long as it will fit in the seat, I say TAKE the car seat! It will be the only way your child will be comfortable enough to sleep, and then you'll be able to relax a bit as well. If your child is in their own seat, they will want to get up, run around, stand on the seat, and terrorize all the adults around you. If your child rides well in the car, she will also ride well in her carseat on the plane. Bringing a car seat, although heavy and awkward, saved my last nerve on many plane flights. I could give snacks, read books, help my kids color, etc, but children knew they couldn't get out.
You can call a skycab once you arrive at the gate, and they can put you and your daughter, with the heavy car seat, on a ride right to baggage.
Another tip, a teaspoon of benadryl. Per my pediatrician, it will help keep her ears clear, and make her just a wee bit tired. BOTH great benefits for her and you!
Happy flying!