Even lead BBs will bounce back. You wouldn't think so, but I've seen it. So use regular steel BBs and eye protection with a low-powered gun like a Daisy Red Rider, and lead pellets with anything more powerful.
I have a moderately-low end break action pellet gun, a RWS-34, and I love it but even adults find it hard to cock at first. A kid wouldn't be able to cock it at all, and might hurt herself trying. I was probably about 8 yo when I got a Benjamin pump-up pellet gun. At 6, I probably could have pumped it up enough but I don't know.
I wonder if Crossman or Gamo makes a beginner break-action rifle with a long barrel that's easy to cock? Or if anyone still makes a pump-up pneumatic?