My son has had no problem killing Whitetails with a 6mm Remington. A 243 will be fine (so would a 6.5 Creedmoor/Grendel/6.5x55). I think a 223 is too small, considering the chance for less than ideal bullet placement, and, though I started with a 30/06, wouldn’t consider a 30 cal if you have the funds to get something more appropriate for a small framed shooter (and you do).

Even with a 6 or 6.5, I wouldn’t make the mistake of going “ultralight” with a kid’s gun. A 7lb or up gun will soak up recoil better, and I doubt your going to have a kid taking offhand shots for their first few deer, so they’ll have a rest to put it on.