Careful when rounding corners, you probably know this, but I just want to mention this for those who don't...rounding an area will allow your hand to grip the stock more comfortably, but that will also result in fingertips now being closer to the string...this is dangerous, so be careful not to over do it.
I really want to sit down with a sketch pad since I like drawing weird stuff, but my eyes are closing on me as I type
....my idea is very close to yours, though I don't know how much detail you are planning on having in this carving, but the wings can definitely wrap around the stock or you can have them folded or you can even stretch things a bit to give it a slightly better fitting and perhaps even a "speedy" look.
By the way, check out this piece by David Swanson...
PS: Just of the top of my head...try searching images of Phoenix on Google.