Actually, the entire purpose of this is that I have a player (my daughter actually) that wants to play an avatar. Not Ang, a different avatar that lives at some other point in the world's history. The idea of the campaign will be to discover this avatar's story. She will be the only player playing a Way of the Four Elements monk.
Your other idea of allowing the Four Elements subclass, but focusing on only one element has also occurred to me for benders in general, but I'd rather let the avatar play that subclass as is, and make other classes and subclasses workable as benders.
And yes, I've thought about how to incorporate the sub-types of bending. Unfortunately, the racial design space is fairly limited. I think custom feats might be good for some of the advanced bending techniques.