I am 65 yrs. old and hunted all my life for mule deer and elk. From my experience, and that of countless others, for muley's, I would not use anything less than a .25 cal. Yes, a .243 will kill deer, but we have lost more deer hit with a ,243 than any other caliber. A .257 Roberts does a good job because the wound channel is larger even though the energy is not substantially more. The .270 Winchester has always been a winner. A 7mm Mag is a lot for most and is quite the meat waster. I have also seen many times where a 7mm mag will blast through a deer so fast it doesn't expand well and just punches a hole.
Since you are looking for less recoil, I think you could not go wrong with a 7-08. The energy is not as high as some of the others, but with the .30 cal. bullet it would do excellent on muley's, and if you ever hunt elk, it would work well if you keep the range to 400 yds. MAX. Elk are very tough animals so shot placement is important but shouldn't be a problem with the low recoil of the 7-08, Muley's are also tougher than most people give them credit for. A large buck can be almost as tough as elk. I have seen them hit hard and still go a long way. I would use a 150 gr. boattail bullet in the 7-08 and you should be good to go.
FYI, For the .270 I have found the 140 gr boattail hornady works fabulously. Not much more recoil than the 130's, but seem to hit a lot harder.