Here is my truck and camper I've never scaled it but the RV manual says 10,400lbs empty and 14,400lbs loaded. With water and propane and gear I am probably at 12,400lbs. My truck is a 01 high output 6 speed, 4.10 gears, RV275 injectors, 60 hp max chip, after market electric gear drive fuel pump, gauges, and a larger exhaust. Around Virginia I tow with my chip in the economy mode and use cruise control in 6th gear. I have not had any problems or issues towing. I have the 265 tires but still need to watch my pin weight. With the camper I am just into my overload springs.
I always add two cycle oil to my fuel at a 200:1 rate and about 100:1 when towing. It is supposed to help lubricate the VP44 pump. So far it has worked for me. It also reduces some of the noise the pump makes.
I think you will do fine, especially since you have a dually. Your truck can carry more pin weight than mine.