I continue to be surprised for god knows what reason when I read knee jerk reactions and dumb comments that you need a DRW truck heavy trailers. Based on your scale weighings the 5th wheel is putting a load on the two rear tires of 5520 lbs. or 2760 lbs. per tire.
Good luck trying to find a E-rated tire with less than 3195 lbs. load capacity. That provides a cushion of 870 lbs. so saying you need two more tires is dumb. Sorry but it truly ignores very simple math.
Easy to find tires from Nitto and Toyo that will work on the stock rims and provide 3750@80 PSI for a total load capacity at the rear axle of 7500 lbs. for an excess load capacity of 1980 lbs.
I have pulled horse trailers weighing over 12,000 lbs. with SRW trucks and I have two friends who each have been towing 13,5000 lb. 5th wheel trailers with their SRW diesel pickups all over the USA.
The absolute need for 4 tires for a 5520 lb. load assumes that each tire has a capacity of only 1380 lbs. which my small boat trailer tires provide.
For grins the next time you take a long trip count the number of 5th wheel travel trailers being towed by a SRW and DRW truck. I have been doing this for the past several years and my tally is always 4 out of 5 are SRW and 1 out of 5 are DRW trucks. Somehow I guess that 4 out of 5 people have figured out that the load on the rear axle can easily be supported by their two E-rated tires.