A DRW has a lower towing capacity but a higher payload capacity than a SRW truck and the higher payload is the deciding factor. After 13,000 lbs. the trailer needs to be a 5th-wheel and that puts a lot more weight on the rear axle. The axle can take the load but not the two tires. Easiest solution is to add 2 more tires to support the load.
More than 80% of the people I see towing 5th-wheel travel trailers on the highway are using SRW trucks. The people I know towing them have a wet trailer weight that is under 14,000 lbs. and they do just fine. One is towing a large 35' trailer and I have a hard time visualizing it as being too small for anyone's needs.