This is why I have and practice with a slingshot/catapult. Have taken rabbits, quail, dove and squirrels. Always eat what you take, remember, theoretically a good catapult is deadly to larger game as well. Be safe, be careful and be smart!
$20 gets you a decent slingshot, 1/4″ – 3/8″ shot , and spare bands. My preference for my bugout and survival bags is using tubular bands, lifespan is much longer versus flat bands, however, you give up fps/power in doing so. But, regardless out too around 30 yards it will still be quite deadly on small game.
I linked the slingshots I use and have tested and will show you, hunting, practice and full reviews shortly. Tapered bands allow a faster shot with up to 3/8″s Inch shot, but NOT MARBLES. Experience may differ and by no means am I a professional, I do however, shoot them well.