I was going through something similar about 10 years ago myself with the oldest grandson. He got it in his red toe head that he wanted to shoot himself a hog. There as no talking him out of it and he simply WOULDN'T shut up about it. I have quite a few rifles but like your issue nothing to fit a tiny kid. Even with my daughter, she started out shooting a Rem 760 in .243 with some reduced loads. Even with that she was still stretching to reach everything.
So I tried everything in my safe and the only thing that would fit the boy was the Ruger Compact in .308 I had been using myself. So I dropped some cash on an EER scope, worked up some of the Hodgdon reduced loads and off to the races we went. He started off with around 36gr of H-4895, shooting 3 rounds at first, then 5, then all I would load up and bring with me. By the time hunting season rolled around the following year he was up to 40 or so grains and shooting around 1.5" at 75 and 100yds. I wasn't going to let him shoot a critter that far out but it made him try harder to keep things where they should be.
I realize the little Ruger's aren't the spit and polished versions of some brands, but they are tough as nails and will hold up to a LOT of punishment. I know mine has been bounced up and off the front end of my 4 wheeler numerous times, run over a couple, and been so full of rain water I may as well have just dunked it in the lake. Wipe it off, spray it down and go hunt.
If you go with the .243, like you mention there are quite a few mid weights that would work well for him even with reduced loads. The first to come to mind is the 95gr Nosler Ballistic Tip. While it might be pretty explosive driven to top end, they behave quite nice in the range of around 2500fps. That would give him a good soft load that would still be capable of driving through the vitals on a deer, if ranges are kept short of course. Then there are some of the lighter bonded core bullets as well as even the Partition in a couple of weights. I wouldn't opt for one of the solid copper ones until you start getting the velocity up into the 27-2800 range for reliable expansion. That might sound high, but with the 85-95gr bullets it isn't really all that.
Give the Hodgdon reduced loads a hard look, as well as their data for the other weights in .243. Any bullet weight that has a listed load using H-4895, can be reduced 60% from the top end. Those make for some nice loads for a youth, especially when they want to shoot, but aren't quite up to full recoil. To be honest those lighter loads in my .308 have been some of the most accurate ones I have used. When we finally went to the full power factory loads, we wee shooting the 130gr TSX over a 42gr load of H4895, which was getting right around 2850's from my short 16.5" barrel. Those are some REAL fun little buggers to shoot as well.
My middle grandson leaned towards wanting to get started early as well. However once he got popped by the recoil he dropped that idea real quick. I got him the Ruger Compact in the laminate stock and it is a good deal heavier than my .308. Even with the reduced loads with 85gr bullets he was backing off. The youngest of the three however is a glutton for punishment. Not to be outdone by "big" brother, he bellied up to the 308 for his first round shot, and hit the bull almost dead center. Course he wanted no more of that action, but he DID fall in behind the .243 and work off about 20 rounds through it. He however hasn't got the strength to manage it though unless it is on bags or a rest. My criteria to hunt is not only being proficient in shooting, but they also have to be able to manage the rifle as well. I can't hold it up for them and aim and have them pull the trigger. IF they can't do those things I might as well do it myself. The oldest, like mentioned above, red toe headed boy that he is, managed the .308 even when he got it over a bit far and worked on his collar bone a few times. He raised up a few tears along the way but, he never said I quite or I don't want too. Now just turning 13 last week, he has shot about every rifles I use, up to and including my Sendero in 7mm STW. It isn't the heavy synthetic either, it is the lighter weight laminate. The scary thing is after he shot it the first time, he had this big ol smile on his face. You know the one, The "its mine you just don't know it yet".
I wish you all the best with your venture. Fit is foremost, and if it fits him good the recoil can be worked around.
