Lightweight .22lr survival type of handgun capable of taking small game? IMHO small isn't the answer, unless you are really intending for it to be a dedicated anti-snake gun using something like the CCI snake shot capsules.
I'd go with something larger, but with an alloy frame. The larger format gives you a longer sight radius which you will need in order to actually be able hit something the size of small game such as rabbits, squirrels, etc.
What I've got is pretty much ideal: A 6" barrel S&W 422. Decent accuracy (more than adequate for small game foraging), and the alloy frame means that it only weighs 21 oz. With an OWB holster it is easy to carry.
Here's an idea: If small game survival foraging is what you are really interested in, you might want to learn how to make snares. Cheap, quiet, weighs next to nothing, no ammo required.