I can't suggest a crossbow or compound bow. I remember how hard it was for me to find a left handed bow when i was a child but I am sure a left handed bow is easier to find in an adult model.
I thought this was pretty cool the first time I saw it. Its called a slingbow, and its a modified slingshot that shoots arrows. They have been used to take everything from fish to wild hogs. If you get one, buy a well made one. My diy attempt worked but broke after a while. Really fun novel primitive weapon that has a lot more punch than a typical slingshot.