M.R.
Call the airline to ask about the carry-on issue. Different airlines do things differently so I would check with your specific airline; this is a question on which I wouldn't want a generic answer. I'm interested to hear that folks cram them under seats because I can't picture a backless booster fitting in the space under an aircraft seat -- We just got back from a couple of flights and though these were huge aircraft, a booster would have to have fit into the overhead as I know it wouldn't have fit under the seats. Since you don't know for sure what aircraft you'll get, check directly with the airline.
He might actually be more comfortable in the aircraft seat if he is actually sitting on the booster just as he does in the car. It will fit -- the airline seatbelts are plenty long enough. A booster will also enable him to better see the TV screen, if the aircraft is one with screens in the backs of the seats in front of you!