Is the 500lb hitch weight the "brochure weight"? Or did you actually weigh the tongue? On average, TW is 12-15% of the total trailer weight. So a 6500lb trailer (for example) could have a TW of 780lbs-975lbs.
LT tires yes, but chances are you will be fine with a LR C. I put a set of Cooper A/T3's on our '10 Sikverado 1500 275/65/18 LR C and never had any issues towing a ~7200lb, ~920lb TW (TW may have been a little more, was 900lbs last time weighed, and rearranged some things for more TW) loaded trailer, with items in the bed and a cab high fiberglass topper.
2 or 4wd? I was very happy with the Cooper A/T3's, in all conditions except never tried them in mud. They got me safely to Northern Michigan and back snowmobiling, as well as the Sault Ste Marie, Ontario area for snowmobiling. No issues in rain or wet roads. Plus a 55k mi tread warranty. Road noise wasn't any more than the stock Bridgestone Duehlers that were on the truck.
It has been since before the BFG A/T KO's came out that I owned a set. Was always happy with them except the price. Main reason I gave the Coopers a try. Save about $300, plus the mileage warranty the BFG's don't have.
Buddy has been happy with a set of Toyo Open Country A/T II Extremes.
Good luck!