Though I agree you can kill small game with a BB or Pellet gun, I don't think I'd pack one for survival. They're too inconsistent and unreliable, in my experience, to be worth the weight. I'd either endure the cost and weight of a powder powered firearm, or, skip it all together. There are other ways to kill game, traps, slingshots, etc, that are potentially more reliable. In addition, for short term survival scenarios, the sort I think many of us prepare for, packing a few thousand calories would easily get you through a few days. Furthermore, you can eat a power bar or what have you even if you're injured, you may not be able to get out and hunt.

Beyond a few days, over which you probably don't need food anyway, the effort that goes into shooting a rabbit or a squirrel may cost more energy and risk more injury, etc, than it is worth. Shooting larger game, where available, would yield more calories.

Of course people will disagree with me, and I don't care. I'm merely stating what I'd do.

On the reliability mentioned, I've stored slingshots, bb guns (pump, spring, and CO2), crossbows, and firearms, for years unused. The only things that consistently works after long term storage in relatively dry space are firearms and spring loaded pellet guns. The CO2 guns inevitably have gasket failures, as do the pumps. The rubber rots of slingshots. The crossbow was a low quality one, maybe a better quality one would last.