OK, my turn to throw out some scattered thoughts.
As has been mentioned, a bottle of multivitamins wouldn't be a bad addition to the medicine kit.
I tend to favor carrying a few fixed and maybe folding knives of moderate size, and using the hatchet or light axe for the heavier stuff, plus a bow-saw or pull-to-cut saw (or just the blades).
A bug-proof tent that one could carry would not be period correct, but it might be really desirable in some locations.
In addition to the snare fixin's, I'd think about bring some net or netting for taking birds, fish and/or small game, unless I could make decent cordage and netting from available "local" materials, wherever we're talking about getting plunked down. I'm not currently familiar enough with steel traps to know what, if any, might be worthwhile bringing in which circumstances.
For reference materials, as well as a good land/water/air trapping reference, a good survival reference for the locale including the most comprehensive possible edible-plants section, because most hunter-gatherers got more calories form gathering than hunting. Of course, one might think about high-grading out the appropriate chapters from several books and discarding any inapplicable bits to save weight. (These are the only sorts of circumstances where I could conscience cutting up books!)
For the firelock in bear country, it'd be a double of reasonable caliber. I agree that avoidance is by far the best policy, but things occasionally go wrong. A double gives a better chance that at least one lock would be working whether or not I'm carrying the right spare parts, and if both are working, then I can keep one barrel loaded with ball, no matter what else I'm doing. I'd be carrying mostly hardened ball with some #5 shot for occasional flock-shooting. I'm not convinced of the utility of buckshot in these situations. The intention is is to hunt as little as possible - remembering that Native Americans snared deer as well as hunting them.
If I had to take one of my current guns, it's be my 16ga double, naturally, but if I could choose a new one, it's be a strongly-built cape gun, with both barrels in 24bore, or possibly 28bore, in a compromise between power and the weight of gun and ammunition. The dimensions of the rifle and smoothbore barrels would be chosen so they would use the same ball/patch/cartridge combinations, and they would be regulated at likely 50yd with a moderate load somewhere in the 2.3/4 - 3.1/2dr range. I'd also try to find a lower-power hunting load that shot to the sights at some usable range, to conserve powder and maybe recover some of these hard balls for recasting or even reuse. The primary sights would be fairly low British express type, with a flip-up leaf with a notch, for the rifle barrel only and at a decent hunting zero, something like 80-110yds, depending on the trajectory of the chosen load. There might be a second leaf for something like 120-150yds, although it seems improbable that that sort of range would be needed or even useful in this scenario (I just like these sorts of things, in addition to just knowing my trajectory and holds for longer ranges).
I doubt that one could stay at one location for three months without straining or exhausting local resources, so I'd expect to have to have the whole kit portable enough to move camp with some frequency, although the rules might allow for caching at the rendezvous/pick-up site, and restocking when needed.
This is far from a comprehensive plan and equipage, but it's as far as I've gotten. Thanks for setting out the scenario. I'm starting to get the hankering for a cape gun of some sort, although the details may wind up different.
Regards,
Joel