Stage 2 and Stage 3 kits should have a full size axe. You don't have to be able to put them in the pack of course, it might be more of a car or carry item. But it's absolutely critical you have a full size axe. The difference between using knife a hatchet and a full size axe is the difference between a .22 a 5.56 and a 308. It also never runs out of ammo
I have a fiskars axe with a composite handle, the lightest I could find. I also use it while camping, my camping gear is my survival gear. To know what gear sucks and doesn't, you need to use it. This will accomplish a lot of things, one you will become proficient with your gear, you will determine the best location for packing it, and you know what stuff you bought is crap and what you never use.
I would also recommend either a high power air rifle 850 fps or better with field points, or a subsonic 22 setup for taking small game quietly. I pack the Henry version of the Ar-7, the U.S. Survival as it weighs less than my silent game taking air rifle, a good old daisy 880 with a scope.
My SHTF pack is a Kelly Bigger Bend 6400. Please don't confuse this with an a assault pack, this is a walk away from the house and don't look back pack. It is designed to carry a full size axe on board, even has a place that holds the blade that is reinforced especially for it and a stretchy strap to hold the handle in place. It is heavy when fully loaded, but I have enough dry cloths and food for one full week and the tools I need to make a permanent back woods residence and defend it.
The rest of the don't look back pack, is my suburban, affectionately known as Grover.
The most important part of any BOB, is a plan, and practice.