The downside of an axe or tomahawk in any melee combat, is that while a blow focused on the edge can be devastating, any blow hard enough to disable an opponent is going to stick into him, and not give you a good recovery or rebound to parry or deliver another blow somewhere else as quickly.
Unless you crush skull into the brain cavity, a baton or club won't be as immediately lethal, but you get better rebound which you can quickly turn into a parry, a block, or another blow.
If you can use the blunt face of a hatchet without cutting yourself, that could be the best of both worlds but IMO, you'd need a ton of practice to be confident in doing it.