Want to get more than 4,000 lbs. per wheel you need to upgrade the rims (which have their own load rating) and the tires to 19.5 ones and pay the $3500 minus what you can get for your factory tires.
I upgraded from the factory tires rated at 3195 to Nitto tires rated at 3750 and my out of pocket cost was $500 net net for the 1110 lb. gain in load capacity at the rear axle.
19.5 tires are going to provide an even rougher ride and the tread choices are limited to highway treads.
Check the overall tire diameter of your current tires and compare them to others and understand that the rim size and tire width are not anywhere near as relevant as the profile numbers in determining the wheel well clearance and the effect on the gearing for the truck. I went from 245/75R to 285/70R tires and the change in diameter was 1.26" and the truck gearing changed by less than 4% which is not enough to impact performance and with a camper load my odometer and speedometer were more accurate with the new tires than with the factory ones.