Kid is 8 or 9? He wants to keep it fun and keep recoil to a minimum. I’d encourage good hearing protection, a stock with good fit, and something with recoil along the lines of a 243/6 creed - max. I’d worry about the “possible” bear or elk later. The “possible “ elk is not so different than the “possible big bears” that so many worry about ... but so few end up pursuing. Choose the cartridge for the predominant role at hand.
Certainly not knocking the reduced load ideas either; but a reduced 7/08 or 308 is still quite a bit more than a reduced 243. My 2 cents.