.243 then......'06
The key point is that the kid WANTS to shoot it. I'd make sure he gets the concept of eye relief w/ the .243, and once comfortable, turn him loose w/ the '06. Odds are he'll be fine, 'cause he's got the desire.
But sensitivity to recoil is a personal thing. My boy, at age 15, can shoot the .270/130, but given a choice, he takes the .243 everytime. My Dad, who was a pretty stout customer; a known scrapper in the community, college football and two black belts in his adult years, did not shoot well 'cause he had a flinch which he did not realize. Dad bought a .243 in his 60's and started shooting and hitting better (w/ a rifle) than he ever had.
One thing I HATE is adults conning or pushing novices into shooting 12 ga, '06 etc, and you see that crap all the time. But this boy wants to try it and I'd let him, once he did the .243 well.