I have a suggestion:
why don't you book a seat for your baby (as a child) and attach the car-seat to the aircraft seat and put the baby in it?

I do this for my son since he was 2 months old and he sleeps well because he's used to the car-seat.
However, I would sugget you to hold your baby during take off and landing.

Don't forget to buy some NEW toys (she's not used to) to entretain her during the flight.

Good luck and have a safe trip!