Yup,.. THIS ^^^A heavier bullet in the 7-08.
Many years ago, I had a gunsmith turn a Weatherby Vanguard .308 into a grandkids rifle. My grandson, who was 10 at the time, was our model for determining how the stock needed to be modified, what brake to use and what type of kick pad. We ended up shortening the stock 1 1/4 inches, installed a Gentry brake and a Limbsaver pad. The piece that was cut off the stock has a Limbsaver pad on it and it can be easily replaced on the stock as the kid grows. I had some 165 gr hand loads to shoot. He was fitted with some good muffs and he shot 20 rounds from a bench, and wanted to shoot more. A friend who was going on a safari and taking his 10 year old son, hear about my Grandkid rifle and asked if his son could try it out. He did and shot very well with it. He shot 5 animals with it and only a zebra required a second shot. My grandson used the rifle until he got big enough to handle a .300 WSM. This rifle has been handed down for over 25 years and is currently being used by my hunting partner's grandson. Having the rifle fit them properly when they are starting out, is in my opinion, as important as the brake or pad.
That would be fun. The OP is looking for minimal recoil, so we were counseling together about some lighter loads instead of heavier that will keep him in comfort zone but still high on performance. That's where the Hammers come in. (wink, wink)I still vote for the 308 win. But i admit I'd like to see what a straight 7-08 with a 180 gr Hybrid or ELDM is capable of. I figure maybe 2550 fps would be common but I wonder if you could get closer to 2700 fps. That would be wicked.