If he is doing so well with the .243 why do you want him to go to a bigger caliber?
The 7mm 06 will only push the same bullet weight as the 243 a couple of hundred FPS faster so there won’t be that much difference in recoil. The big thing would be to stay with lighter bullets.
243 100 GR Max velocity = 3100
7mm 08 100 GR Max velocity = 3277
This is data taken from HODGDON web and only shows the max velocity achieved with a 100 GR bullet for each.
http://data.hodgdon.com/cartridge_load.asp
Has he shot bigger guns?
I would think that gun shyness shouldn’t be that big of a problem if he has taken that much game already. Find something he can try and let him make the decision. To me the sharpness of the recoil is a lot worse than the energy. I have shot thousands of 30-06 with steel but plated military rifle's over the past 30 years and recoil never bothered me once.
A friend last year asked me to help him with hunting loads for a 7mm Rem MAG. This was a light weight hunting rifle. 3 rounds were enough for me, I hurt for 2 days. Then again the gun didn’t fit me worth a hoot.
That brings up another point. If he is not that big, having a gun that is too long will make the recoil, any recoil worse.
What did you do for length on the 243?
Personally I think that it’s really neat that you have a 10 year old that is that gun savvy to have taking that many deer.
Sounds like you have made a good father son combination. Good for you.