I would suggest a .308 sporter and just whack a couple of inches off the barrel and stock if you want shorter.
I cut one down to a 16" barreled .308 with several inches off the stock as well a while back and love the way it handles now. I was reluctant at first, but after seeing what guys are doing with the 12" and 14" barreled striker handguns I ahead and I'm glad I did.
As far as lighter recoil I load 38gr H4895 and 130gr TTSX for the younger kids in the family to shoot with less recoil than a .243.
Using that combination you can tune the loads practically as light or as stout as you want within reason.
The .308 loaded with the 130TTSX is capable of some pretty impressive ballistics and seems to shoot much better than other light bullets I've tried in .30 cal.
Good luck and thanks for your service.