Don't know how common it is or how hard it'd be to find a barrel, but I swear some time ago when I was searching for a .256 Win.Mag. barrel, that someone mentioned having the same type round except made by necking down the .357 Maximum (.256 Maximum?) Of course w/this round you couldn't get up to the 130 grn bullet weight, but probably 120 -125 grn would be possible.
Of course w/the real mild recoil you are looking for the 6.8 w/the 115 grn load should work well. Then on the other hand SKB2706's idea of a 7-30 Waters would be great, I know guys with 14" barrels that claim the 120 Ballistic Tip is just wonderful on deer!! Personally I've had a 6.5JDJ 14" barrel that would shoot 120 BT's @ 2400 fps+, that was pleasent to shoot, I imagine that a 16" - 18 1/2" carbine would be quite mild in recoil.
Well, best of luck w/your choice.