I was just going to suggest this. You beat me to it!

A slingshot with hunting-power bands on it will have a lot more power than a pellet gun. However, a pellet gun will be easier to shoot accurately at longer distances. I would say a slingshot would be good out to 20 yards for a practiced, but not expert, shooter. Much farther distances are within reach with more practice and skill development.

A BB gun - I'm not convinced that this will have anywhere near the power you would want.

Below is a good setup to start out with. High quality. Inexpensive. Indestructable. Use the 4-strand tubing set for practice, the 8-strand for hunting.

http://www.dankung.com/emart/jungle-hunter-dankung-slingshot-iiusalong-handle-p-319.html

http://www.dankung.com/emart/rubber-tubing-set4strand-1745-p-250.html

http://www.dankung.com/emart/rubber-tubing-set8strand-2040-p-232.html

Then go to http://www.slingshotforum.com/ to learn about shooting and how to develope your skills. Practice shooting with GobStoppers jawbreaker candy (cheap, and available at your grocery store). For hunting, you can buy steel or lead shot. Or just find some good rocks or heavy nuts from your garage screw/nut/washer collection.

p.s. - The heavy band sets are for shooting heavy hunting ammo. They won't make a light projectile (like GobStoppers candy) go any faster or farther than a light set of bands. So practice your shooting to gain accuracy with light bands.