Many prefer a .22 lr for a long term game getter. Keep in mind, one .357 cartridge weighs about as much as five .22 lr bullets. The 22. is light and it's accurate. You can buy higher velocity CCI Velociter hollow point ammo for small game. Deer have been taken with this ammo as well.
While there are many break down 22 rifles, for storage in a long term BOB, I prefer a .22 lr handgun made by Hammerli a Swiss company, for Sig, it's called the Trailside (in Europe it's the Xesse) and is, by far, the lightest, most accurate, and most precisely made of any of the .22 handguns (6" barrel preferred). These have a 3/8 dovetail on the top for optical sights, and the target model has adjutable iron sights. While the gun has been discontinued, it can still be found for $400 to $650.
The most important thing about owning a gun, is knowing how to care for it, and how and when to use it.