Frankly I would just try the tethered thing. Give it a test run and see how much you really snag or not. you can alway walk around untethered and only tether when needed? One gun that is very promising in this kind of use would be the RTIs (Priest, etc)
Depending on your needs/wants a good springer is always an option as long as you can take down what your quarry is, albeit not sure that it's backpack friendly.
Anyways, I think you have options, just have to figure out what's more important to you and your specific needs.
As for the kind of gun, caliber, shots needed, etc, If you are using a 357, you must be after something a little bigger than a squirrel. Pigs? Rotes? Deer? How many animals are you realistically looking to take down? (including potential missed shots) Can't be that many. If shooting anything from a Fox down, I recommend a smaller caliber gun, in which case you don't really even need backup air. My .22 Cricket Mini can take a Fox, no problem, and take another 56+ shots after that. Otherwise the information is inconsequential.