I built sort of what you're after last year; Rem M7 in .260 Rem with a Manners stock and fluted 20" 2B profiled barrel.
I really like the combination of 20" barrel with short action for stands and blinds, but still has the trajectory to cover a row crop field (Eastern KS row crop field) if needed. I scoped it with a somewhat compact Leupold 2.5-8X36, for a little over 7lbs "all up".
If you hate recoil the 6.5s are a good option, I get 2930FPs with 120 BTs and 2850 with 130ABs. I started my son out with 100grn BTs at 2850 which worked great at woods distances and felt like a .243 low recoil load. For the distances I'm using the carbines for both loads are adequate. IF I were going for the "all around" and including the possibility of elk, I'd go up to a 7-08.
One thing to consider is that while 20" barrels in the caliber you're looking for might not be that common, having a longer barrel cut down and re-crowned isn't all that expensive. I know Ruger makes some compact guns 20" as do Christianson, and Savage.