if your welding is OK then welding it up wouldn't be a problem at all. It could be made to look OK too, but never as new. Depending on how beefy your welder is, you will either need to grind a deep V to get a root bead, or just grind away the surface crack into a shallow V to get clean metal. Wrap the edge in a damp cloth to protect the temper. Preheat the weld area a bit with a blowtorch and then run several beads along the crack. The subsequent beads should help to normalise the HAZ from the weld. It's unlikely that you would ever break it again
People get too snobby about electrically welding tools and if done properly are just as strong as if welded 'properly'. Not that I do it in my makings, but if it were my own axe head to be used, then I would have no problem doing a quick and dirty weld on it!