My personal opinion, I would not be out there with anything less than a .243. Obviously a .223 can kill a deer, but if the shot is not optimal, the odds are much lower than with a .243 or 7mm-08.
Is the 7mm-08 a barrel? If so...put that bad boy on and let him shoot it and see how he does. He should have no issues at all handling that, he just needs to learn how to hold and shoot it properly. Let him practice enough to gain confidence. If you are having scope issues and he has to creep way up, you may need to get a longer eye relief scope for it. Good luck man, nothing funner than watching your kid get his first.