I've had the Cooper Discoverer A/T in the 315/75/16 on my 01 2500 Shortbed 4x4 for over 50,000 miles. They still have plenty of tread left, but I'll probably need to replace them before next winter. Traction is a thousand times better than the factory Michelins (which I changed at around 50,000 miles with still good tread left). Granted, the size makes a big difference, but with the Coopers I actually drove 55 miles of mixed city/highway snow covered roads to work all in 2WD. With the michelins, I couldn't get up my driveway in 4WD.
My only complaint is the load rating "D". With the factory tires, I didn't have any steering wander, but with the coopers it is very noticable. I have hauled 2,500# of dirt in the back of my truck with both sets of tires. With the factory tires, you could hardly tell it was there, except for the fact that the suspension actually worked in the back and the slightly longer stopping distance. With the Coopers, it felt like it had horrible body roll, but in fact it was the squishy tires. In both instances, I ran the tires at max pressure (50? psi cold for the coopers vs 85? for the michelins).
Overall, I love the way the bigger tires look, as well as the increased traction, but I wish they made an E rated 315/75/16. The only E rated 315/75 I can find in any brand is on a 17+" wheel. I never tow and very rarely haul anything, but if I could find E rated tires in the right size I would switch, but I wouldn't count that against the coopers, just against the D rating. I couldn't be happier with the traction, road noise, tread wear, and price of the tires.
As far as rubbing, I get the typical lower control arm rub at near full lock, which was unaffected by the leveling springs.
All that said, I do recognize there may be a difference between the ATRs you are considering and the A/Ts that I have (such as tread depth), but I imagine the driving characteristics for the same size D rated tires are very similar.