I have never seen any of Hansens work and so I did a search and all I can say is "WOW". Those are some of the prettiest hawks I have EVER seen. I think for the functionality you are referring to I would go with the belt ax. Some of those hawks are so pretty I would hang them up and never use them.
Pros of the ax would be its "generic-ness", the fact that it would be easily replaceable. In addition, the blade geometry is better setup for the kinds of outdoor duties you described. The backside of an ax makes an excellent tool for putting in tent stakes as well, although a good poll on a hawk can do the same thing. The handles would also be easily replaceable assuming you broke it.
Pros of the hawk (depending which one you were looking at) would be increased range, possibly more power, blade geometry set up for fighting or cutting (angles are different) I like my hawks SHARP SHARP SHARP, and Ive actually cut myself with them as easy as I would with a knife. The disadvantage of that razor sharp blade is that the edge cannot handle chopping as well as the ax. In addition, those are some very good looking handles, and I'd HATE to break one. It would of course, be easier in my opinion to replace a hawk handle to an ax handle. The back side is either spiked, hammer polled or plain, spikes have good uses (more than "sticking through a helmet). I use the backside of my hawk like one would a climbing ax on steep ground, and it helps out.
I guess my conclusion would be to go with the ax. They are less versatile, but for the specific work you've described better, although the ax and the hawk both have pro's n cons. I'd just be scared to really use and abuse a nice hawk like that. This doesnt mean I wouldnt want to own a nice hawk like that though