AA missiles aren't any epitome of destructive power. They don't need to be since our airplanes are made of aluminium, composites, and other light but rather weak materials, combined with the fact the vehicle effectively further destroys itself when it's trying to fly while damaged. I doubt they would really do much more than piss off this dragon, which could stand multiple hits from the RPG HEAT warheads. They would need to throw something like Hellfire at it. Or whatever the Japanese use. It would probably require the dragon to be landed or at least hovering.