The gentleman said he can't take much recoil due to his back injuries. Obviouslywe don't "feel" recoil when taking a shot at game. We've all experienced that. But, the recoil still happens. The jolt of some of the heavier cartridges listed doesn't change just because we're shooting at game rather than at targets. Not knowing what his injuries are, it's entirely possible that the recoilmight cause further injury.

I'd go with the 7mm-08 in a heavier rifle with some sort of recoil dampener. The 6mm will work with the right bullet, limited range, and perfect shot placement. But, they don't always stand broadside at 50 yards...