Suggestions are usually too subjective (what did he say
), most often when you ask what's the best what ever, you'll get a variety of answers that likely don't include much thought about your specific needs but rather reflects more on the favorite whatever of the person offering the suggestion.
My answer to your question will not be much different, you mentioned three clues for your need for carrying a hand axe, the first, most of your woods bumming is light weight back packing, but you're not a distance hiker, you expect to use it to process wood, I'm assuming you're talking about firewood and possibly some light camp projects, and you intend to use it for carving, these clues should point you in the direction of a reasonably small axe, say with a head weight of about 1-1/4#, and a handle length of about 12"~ 18 inches.
Of course it also depends on the area you'll be hiking into, is it mostly hardwoods or softwoods, maybe a mix of both, how's the availability of firewood and carving wood, is it easy to get or are you going to have to work for it (process big logs into more manageable pieces).
If your somewhere in the middle then perhaps a heavier weight hand axe would be a bit better, one with a head weight of 1-1/4#~1-3/4 pounds on a handle about 18"~ 20 inches long and able to use two hands if needed.
With a budget of about $150.00 you are not limiting yourself to hardware store budget minded axes, that's a good thing, most mid priced brands like Huskies, Snow & Neely, and Council Tool are a pretty good compromise between quality and price, I've tried most of them and found them to be ok "if" you aren't particularly fussy about balance and handle size, I am one of those fussy kind, if it's not comfortable to use (even briefly) it's not for me, I've pretty much settled on a few hatchet/axes that fit me perfectly, two are Gransfors Bruks, the Wildlife axe and the Small Forest axe, which one I carry is dependent on my planed needs for that trip, my third favorite is a Hunter's axe I put together from a vintage Swedish forged head weighing about 1-3/4 pounds on a premium quality 24" figured hickory handle for those bigger camp chores.
I have heavier weight axes on longer handles of course but my lumberjack days are way behind me, so, my suggestions are don't skimp on quality, actually handle the axe before buying it to make sure it fits you, don't buy more axe than you need size wise, a good quality well balanced axes can usually be counted on to cut well way above it's weight class, and don't discount vintage tools when you can find them in good shape, refurbishing and rehandling an axe can be fun and rewarding, there are quite a few gems still waiting to be found at yard sales, flee markets, and junk shops, and there are several good handle makers still left around.
Good luck with your search.
