Grow the handle through the axe eye...

[kvaughn]

50+ years ago while deer hunting on the Brazos River in north central Texas I found a break down single shot shotgun action and barrel grown into a hackberry tree. Some remnants of fence wire were still there. The stock was long gone and and the tree was about 10" diameter. Hackberries grow small and spindly.
It was on an obvious game trail crossing a small creek. I figured it was a game trap somebody couldn't get back to or forgot.
I always wanted to go back with a saw but never got around to it.
Kind'a off topic. Went down the rabbit hole with trees growing around things. Sorry

 

[kvaughn]

Im thinking this might possibly work if you had a smaller branch from a mature tree growing through an axe head. Maybe Osage Orange, its got a lot of shock resistance.
Put your axe head on the end of the branch and slide it down as snug as you can and walk away for 2-3 years. Come back and hack off the branch, shave it down. Maybe it will have grown around the head a bit at the eye are

I think Osage Orange grows slow. Would be tough though.