Welcome to the forum. I have a 2004.5 3500 truck except it is a 2WD dually. It is pretty much the same truck except 10 ft-lbs less torque on mine assuming you have the 5.9 Cummins. My numbers are 21,000# GCVWR and a tow rating of 13,900#. Those numbers can be bumped 2,000# with the 4:10 rear end. These numbers say no to your trailer. However, it wouldn't bother me a bit to hook up and haul your trailer in an emergency.
I have a buddy I camp with a lot that has a 2006 2500 that pulls a large Montana most likely heavier than your trailer. His truck is lifted slightly with oversize tires and a lot of tuner goodies including a noisy 5" turbo back straight pipe and air bags. He has been towing his rig with that truck for seven years now that I know of with no chassis problems. However, he is on his third transmission with the last two being HD built with all the billet goodies and triple plate TC.
I tow 10k# to 11k# with my truck and it is a comfortable, easy tow. MY exhaust gas temperatures will rise on long steep grades. The transmission will get hot around town and on climbs slow enough to keep the torque converter out of lockup. If you have the diesel, an exhaust brake is mandatory. Did I say an exhaust brake is mandatory?
Bottom line, these are stout reliable trucks. The 48RE transmission is a weak link and is most likely a reason for the low tow ratings. I would like to have extra gears, but the 4 speed works and I have had zero issues with mine. I have about 50,000 miles on mine with the trailer hooked up.