If you want to perform as an axe, then buy an axe, and carry (also) a folding knife.
A "good" knife, capable of reasonably heavy-duty...but NOT as an ax..
Spyderco Gayle Bradley...3.5" blade of M4 steel. A great folder.
Spyderco TUFF...4" blade of CPM-3V blade steel. Also great...even heavier-duty than the Gayle Bradley.
Spyderco Military, available in various blade-steels
Benchmade Adamas, model 0275, in D2 steel, 4" blade...VERY strong, capable knife, will take lots of abuse.
All of the above are pretty cheap for you, from about $150 USD to $220 USD.
I would also suggest that you look at Zero Tolerance folders. They are made for heavy-duty, and many of their models might fit your needs.