I'd stick with a sporter type barrel that way the gun is versatile and one that's handy to pack around. 22" with a muzzle diameter no heavier than .620, itll still be muzzle heavy enough to settle fast but not so tippy that it needs a heavy stock to balance it. I have a Ruger American Predator with muzzle diameter. 750ish and it's too much of a good thing in that way.
250 Sav has been called "the closest thing to a recoiless hunting cartridge ever invented" forget who said that but I agree. Ive killed a bunch of deer with em and my dads killed a whole bunch more and a bunch of hogs. I wouldn't hesitate on elk at normal ranges just know the bullet and its limitations and workin with it. Not sayin it's ideal for that, jus sayin itll get em.
It kills really well with normal bullets and can be twisted 1:7 to handle heavier bullets if wanted ie 131 ACE, but itll kick less with ur 100ish grain Hammers you mentioned. 1:8 will get you to the 121s and still be good with 90 grainers I been told. Blackjack bullets shot a 1:7 and 100 grainers still held up n shot in a 25-06 if I remember correctly.
Other good choices 25 Creedmoor=same as 250 Imp, 250 AI is ballistically the same just diff shoulder angle n ..
243, 6 Creed, 6mm with 1:8 to handle Hammers and the heavier ELDXs n such.
The 6.5Creed, 260, 7-08 are awesome. But they do kick noticeably harder than those above. Ive had a 7-08 and have 2 Creeds, they for sure kick more like a 308 than a 22-250. At least with 140-147 grain bullets. You can load down, lighter 120 grain bullets to more moderate speeds and be fine tho. I wouldn't go above this level for sure.
My nephews are 13, 14 and 18 now but I have had them shooting with me for years. And that's what helped form my opinion on this.
Oh yeah I wouldn't brake it, for muzzle blast reasons made worse by shorter barrel as well as the potential for hearing loss if theres ever quick shots hunting n ear plugs werent in. For me I also dont use em because I hunt with horses and a dog quite often n they dont have hearing protection.