I am a firm believer in setting kids up with a fairly heavy (for them) rifle, recoil is far less of a concern than a light rifle of the same chambering. Heavier rifles are more stable, don't move as much if the trigger gets yanked in the moment, etc. All around easier to shoot rifles. Increase the chances of them being successful.
If elk are on the menu I would personally go with a 7-08 or a 308, it is my opinion that 243 and elk is an expert level combination regardless of what bullet is used and I don't expect kids to have that level of marksmanship or understanding at all. Even if one is an expert I don't see why you would pick a 243 for elk unless there was no other choice? Yes I know it's done all the time and no that doesn't change my view one bit. A 6.5 with a good bullet would be better but I would still like to have 7mm or a 30 cal even better if this particular kid can manage it. Again, increase their chance of being successful. I think the 243 is an EXCELLENT cartridge and I bought one for my kids starting out too, but southern whitetail deer were the biggest and toughest animals considered in that decision. I would argue that 243 might be the best choice for southern whitetails even. My son was strong enough and had shot enough that he started out hunting with a heavy 308 and never hunted with the 243 at all, the heavy 308 had much less violent recoil as an added bonus and he not only shot it better he preferred to shoot it. Small light 243's can be sporty recoiling rifles for kids.
If any way possible I would recommend a suppressor. I wouldn't recommend a brake unless you will be able to use double hearing protection for the shot AND the kid has shot that braked rifle enough to be used to the concussion.