You don't need a large warhead.
With modern infrared or radar seekers, you don't need a warhead period. It would probably be more lethal against large aircraft if it were hit to kill rather than a warhead, or some sort of SAPHE warhead that detonates inside the aircraft structure, but no warhead is probably preferable to a tiny one in practice. The GD AAAM is big enough to be a rather muscular metal rod that punches a fat hole in an airframe.
Planes generally dislike having big holes in them, you can probably carry as many or more of these little missiles as AIM-120, and there are only a few dozen functional Backfires left in the world.