Ok, time for me to get up on my stump.
Buy the .243, definitely, but for yourself, not for your kid. The absolute worst thing you can do to a new hunter, kid, wife, whatever, is to saddle them with a marginal cartridge on their first hunt merely because "it kicks less." By that logic we all should use a .22 short. Ok, .22 CB cap.
Wrong. Way better to give them a slightly heavier gun in a way more adequate caliber. You want their first deer to go down in a pile without a twitch, not flop, squirm, and scream for 20 minutes, otherwise you're taking a big risk on turning them off hunting, or even to anti-hunting, the very first time out.
I'd suggest a 7mm-08. Works _good_, makes deer flat without kicking too much. .270 or .260 if that's what you've got. Even the .25s, which I love, are not wise choices for beginners.
Buy that .243 though ... for yourself. Or use the action to build that 7mm-08.