Leave the AP cost as it is, but make the damage 5-8 instead of 1-2. This would make the BBgun a tempting alternative to 10mm pistol for newbs because 10mm ammo is hard to make.
It's a bb gun, the real life equivalent you'd be hard pressed to kill a small bird at ten feet.
I don't know what BB guns you've been using, but my cheap run-of-the-mill pistol can shatter bottles from 10 metres. A reasonably well made rifle can easily go right through a small animal from 10-15 metres. With pellets you'd probably kill any small bird from 20 metres if you can hit it.
You're forgetting how old and shitty the red ryder bb gun is, it's a toy.