Ok here is my thoughts. If you place WTB post in exchange asking specifically for rapid fire target pistol, you may be amazed at how many are still circulating. They became pretty much useless for any competitive shooter, although I would guess all competitive shooters used to have at least one.
Some traders here on CGN used to carry them not too long ago for cheap, I am talking literally couple hundred bucks, nice guns, target grips and all.
And lastly, everything you shoot out of pistol length barrel will be very loud, your kid will still need a hearing protection. But then, maybe if your kid shot 22lr and then 40SW or even 9mm and then switched back to 22, this maybe a better way to build shot sensation tolerance. JMHO