I really enjoyed my H&R Handi Rifle in .300 Blackout. Single shot and my suppressor/silencer worked like a charm. I had the AAC version, they are getting a little hard to find but worth the money.
.243, pretty soft recoil, 100-105grain bullet good for varmints, too. Old hunting buddy of mine starting using a Ruger bolt action chambered for .243 about 35 years ago, started hitting coyotes with it out on his cattle ranch so effortlessly he felt like it was nearly magic. Then whitetail deer were harvested every year since. With a $100.00 Bushnell scope on it.
My first question would be what ammo have you been feeding the .223? With solid copper barnes 62gr we have consistently had ( short) blood trails. They nearly always exit. ( we’ve had one out of eight that didnt exit, and it was a frontal/ lengthwise shot)
I wouldn’t worry about a .243 being too much gun for your son. Now I say this only for reference, my daughter is 100lbs soaking wet and doesn’t have a single issue with a .243. It’s a great round and he should enjoy the step up.
I've got a Model of 1917 in 375 H&H. It's absolutely perfect for a teen. Reasons:
1. After you get used to shooting this, recoil won't be felt for the rest of your life.
2. Cost of ammo with cause one of two things:
2a - You'll think about your shots a lot more before you put your booger finger on the bang bang switch
2b - You'll learn to reload and have a wonder life skill.
3. You'll be able to take almost anything in the world. You'll never NEED another rifle.
4. Bragging rights
5. I'll make you a great deal