FWIW. I've thought about this because my choice for heavy cutting is a hawk also. The problem is always how to carry, the handle being the real problem. The modern hawks are made to slip off the handle, slip on and tighten with use. That being the case, after cutting some branches with it I'd mark the handle for actual using length and cut off the excess. Then I'd carry the head in a sheath in my pouch, form a metal ring around the upper part of the handle that it will not slip all the way through. Mount the ring on the sholder bag strap, slide the handle into it and tie a loop that is affixed to the bag strap around the lower part of the handle to keep it in line with the strap and in place. The handle can be hung this way from the front or back of the strap. Just a few seconds and it's assembled and ready for use. If you don't expect to meet any bears face to face this should work.
But I will say it's not how I carry my own. My hawk is homemade from a black iron pipe Merchant coupling with a blade made of 1095 steel from a plow disk welded to it. (Yes, I know people will say the weld can fail, but I use them hard and have never broken one.)I don't carry a handle at all. When I need one I use the head as a hand axe and cut one. drive the head into a stump or dead log and screw the handle into place. If I want a better one I can now make it with the first one. I just carry the head in a sheath. Simple, and works for me.