Personally, I've never found a knife suitable for both batoning and cleaning small game. But then I don't ask a knife to prep my firewood. That is a job for a hatchet, axe, or saw IMO.
I do own many hunting knives, that work at the campsite as well as in the field. Many available for under $100. I tend to gravitate towards the drop point. I've always liked the way Boker does it. My Relincho is good though, at a 5" blade a bit bigger than I normally like. I prefer more the size of my El Trampero or Campanero though those are discontinued. Boker's Pine Creek, Dano, & Hunter are similar and available.
I've carried my Buck 102 woodsman for decades in the field, though it is smaller than what you seem to be looking for.
I have a Browning Overtime I like, though the blade is under 4" and closer to a B&T knife. If you look at Browning, AVOID the new Ignite. It is right what you are looking for being a bushcraft / hunting knife and only $50, but they currently have a defect in the bolster that will IMO significantly weaken the knife.
If you are willing to spend a bit more, I like Benchmade's Saddle Mountain Hunter or Skinner. Also check out Enzo's Badger or Trapper 95. Bradford's Guardian 4 is also well regarded.