Good guess, but not what I have in mind. So far as I know, this one had a major role in only one movie, though it may have appeared occasionally in others. Here's the additional clue for your guess (though it would fit the Tiger Moth, too).
Looking for the name of a warbird.
1. Played a prominent role in a successful and still highly regarded motion picture.
2. Single piston engine.
3. Crew of one or two.
4. Fixed landing gear.