Nicholas, one thing to consider there is the thickness of your saw blade. I own a small hunter's hatchet, from Western. It's a flat 3.16" thick, and around 11" over all length. It's small, and very light compared to a hatchet, or even other traditional hawks(with an eye through the head). I wanted it for backpacking, thought I could use it to make kindling etc. But, it sucked at actually chopping, due to it's lightweight.
I would like to make a integral hawk from a thin stock, but not for chopping or chores. I'm thinking more for a self defense weapon, lightweight, very sharp, and very fast in hand. It would lack the mass to have much momentum, but it may slice well, IDK. I'd like to try out Pipeman's little Winkler -Sayoc style hawk to get an idea of it's feel, or just make my own. I'm thinking concealed carry would be perfect for this type of hawk.