I have a pretty head strong grandson. When he posed to me he wanted to shoot his own boar hog, mid spring several months before his fifth birthday, I figured it would pass, but it didn't. He insistantly brought it up every time we were together and even on the phone when we talked he brought it up.
Well I went through everything I had and the only thing I had which he could possibly hold up on a rest, on his own, was a Ruger Compact in .308. I pulled the data from the Youth Loads on Hodgdon's site and away we went. Took a few times to work up to more than 3-5 rounds at a sitting, but he was determined to get his hog, what ever it took.
This was mid -Oct. three years ago, with him setting up to shoot at 50yds,
His results that day,
Then the first weekend of that November two weeks before his birthday,
If you can find one of the Compact's I highly recommend them. I was shooting this particular one very comfortably for a couple of years before he swiped it from me. The reduced starting loads using 125 - 130gr bullets are less than that of a .243, and will do the job out to 50 - 75yds easily, and using the H4895, you can continue to add to the load all the way up to full power for caliber with out changing powders. We started out at the bottom load and within the next year had worked up to 42grs, and switched to the 130gr Barnes TSX and now the TTSX. This particular load averages just over 2650 from the stubby little Ruger and he can shoot groups under 1" at 100yds with it. IT has plenty of punch to reliably drop up to 200# feral hogs with complete shoulder shot pass through. The recoil seems to be just now getting up to a standard .243 factory load level, when I compare them side by side. However the .243 I used has a 24" barrel and weighs a couple pounds more. The Ruger as it is in the picture weighs just under 6.5# with a loaded magazine. the kid is now 6 and has much better control of it and has taken another hog and a coyote at 158yds with it.
I wish you luck in your quest, as I know from experience that the time ahead of you is very rewarding. Just remember that patients is the key and a few rounds at a time is better than them learning to relate shooting with a sore shoulder. I got my first deer when I was six, and then I faced these same issues with my daughter when she was 6, so I guess it runs in the family.
Again enjoy the time spent they grow up WAY to fast.