I've checked around with a few airlines. American Airlines is typical -- they will not accept for flight passengers "unable to sit upright with seatbelt fastened. The only exception is a passenger in a body cast. Contact American Airlines Reservations for details."

But I wonder just what they DO for a passenger in a body cast. They won't accept passengers on stretchers, for instance.

They mention a few special seats reserved for passengers with disabilities; as best I can make out the main advantage is a moveable aisle armrest so you can slide in to the seat.

United similarly mentions moveable armrests.

Some airlines will charge a lower fare (75% or 50%) for a second seat if you require one due to your size, similarly to how they will charge for a cello.

Nobody mentions upgrades to a premium cabin if your doctor says you need to lie down during the flight -- mostly they suggest then they won't take you at all.