I load a 7mm 08 for the kids with about 36.5 grains H4895 and a 120 grain bullet. Barely more recoil than a 243. You could probably get close with the 125 Noslers in 308 and about the same charge, but I haven't tried it.
BTW, out of a Bergara carbine barrel, Norma brass, turned necks, BR primers, proper headspacing, Hornady 120s, I get 6s to 8s most of the time out of the 7mm 08. It is rare indeed that I shoot a group over an inch at 100 yards, with 180 rounds through the barrel. Working on seating depth a little, thinking I can drop into the 5s more often, but then I'm done. IMHO, that's gonna have to be good enough for an 11 year old to take a whitetail.]
I love that barrel.