You started off great in your post. It's a shame everyone confused you.
You have to realize there are two different methods of calculating what a truck is capable of. Some people look at the GVWR and have a fit if you dare go over it. They maintain that the GVWR is the absolute maximum a truck should ever carry.
Other people do what you started out doing: you looked at your truck's tire rating and axle weight to determine how much extra capacity you had for pin weight. This will almost always result in a bigger number than going by the GVWR. This method is the one that the professional trucking industry typically goes by: tire weights and axle weights.
The rear axle in your truck is an 11.5" made by American Axle Manufacturing. They rate this axle assembly at 10,000 pounds. Obviously on your truck the tires are the limiting factor. Your could actually upgrade to 19.5" wheels and tires and be able to realize almost all of your axle capacity.
As to the question of "will you be pushing it?" Yes, you will be.
Your truck is marginal for the trailer you're looking at. If you change your tires to heavier rated ones and possibly add some air bags or heavier springs you could successfully tow that trailer. Would you be over the GVWR? Yes. Does that bother you? Only you know.
My personal rig is a truck camper on a '05 Chevy dually. Like many TC owners I'm well over my GVWR of 11,400 pounds. I've logged many thousands of trouble free miles at about 13,100 pounds on this truck. I'm not over my tire, axle, or even tagged weight so all is perfectly legal.