M.B.
Really, really, really I would try to get a seat for her as ISTM based on your description that she will be screaming during the flight - she will have to stay on your lap for well over half the flight. The least you should do is check with the agents when you get to the airport to see if it isn't possible to get an empty seat next to yours. And, then just plan on gate checking your car seat if there isn't an empty seat. If you happen to be breastfeeding then it should actually be easy - just get a window seat and nurse during take-off and landing. I did that with my sons and often my younger son would then sleep through most of the flight. Barring that, if she will at least take a bottle while you hold her, that might help, in particular she really needs to be drinking something at takeoff and landing so her ears will pop, otherwise, she may well be screaming in pain. Also, I would bring any soothies, cuddly things she uses at home at bedtime to keep her more calm.
And, one final thought is to give her benadryl as soon as you board the plane. That might help her to sleep and benadryl is actually very safe for kids this age. Personally, I wish more parents would do this as I have endured way too many flights with screaming children. And, I am not one to give my kids any medications unless they are absolutely necessary, but this seems like such a situation to me ;)