I recommend buying one of those carseat bags/backpacks and then reinforcing the seams with duck tape and using that!
How old is the child that will be using the carseat and will they have a seat on the plane?
I had carseat, stroller, 2 backpacks and 2 kids for our last flight and while its challenging, its doable and was totally worth it to be able to strap one kiddo down for the trip!!!