For 2-300 yards and working out of a short barrel, 6.8 SPC II might be something to look at if you can get that bolt face. I have box ammo loaded with Barnes solids and that stuff works.
I don't know anything about 6XC but I imagine boxed hunting ammo is harder to find.
If you have to go with a standard bolt face, 308 isn't exotic but it will fit, you can get box ammo with non lead bullets and 308 will let you expand the range a bit.
The only reason against it might be bullet cost if you are reloading but I haven't priced 6mm vs 6.8mm vs 308 Barnes bullets.
7mm08 might be very good too.
I know 308 is a pretty good short barrel option but if you reduce the bore you might end up essentially wasting powder.