I like to take one if my hawks along while hunting or camping too, I try not to get caught up in the arguments about how a big knife or an axe would be better, nor do I care much about weight because the majority of my travel is done on an ATV and I'm moving so slow while stalking that the weight doesn't matter much at all ( my rifle is 11 1/2 pounds ).
I've had a Shrike for a while and am amazed at how well it works on wood, it moves pretty good chips and it bites pretty deep. I have quite a few other traditional hawks and they also do well but they have much heavier heads and longer handles; since I'm not a kydex bender non of them are as convenient to carry. I really like the Shrike a lot, I don't find it to be too long for carry, just right for use, and a whole lot of fun.
I read some good posts by Mistwalker about using the Shrike as a woods hawk before I bought mine, try looking them up, he did a great job.
I bought a Jenny Wren last Oct. it's very easy to carry and convenient to have around since it's so light and short that I never take it off when on a trip. I've always liked big knives but rarely carry them in the woods, the JW fills that role for me. It chops well enough from my experience on firewood and field dressing a white tail deer. It's definitely no axe, nor was it meant to be a full size axe, but it handles the firewood well enough and you can baton the handle if it gets stuck. I would never go any smaller than the JW for wood work and honestly it may be too small for most, but it's fine to me. I'm not out there depending on the quickest fire or the biggest tree, I have plenty of time to gather my firewood and don't need the biggest fire since it's mostly for pleasure and not survival. If I was out there in that fashion I'd have something bigger but the chances of me getting lost doing what I do are astronomically slim and I get cell service there so if worst comes to worst I don't need the bigger axe.
All that said, I'd have no problem taking either one with me on any hunt or adventure, even if it was a remote adventure where a big axe would be necessary- I'd still be wearing the Wren, it's just awesome !
I remember reading a post not too long ago that RMJ was going to be releasing a dedicated woods style hawk, then the newest CRKT collab hawks came out, you may want to look into them.
Good luck choosing and please let us know how you like whichever one that you end up with.