You didn't get a bad number, I'll bet. Weights just add up faster than you think. But it is kind of silly that they could not give you axle weights. I would run over a scale again, and be sure to get that rear axle weight. That one is key.
With the fivers you hope to pull, you're going to have to accept going over the GVWR of your truck. Not a big deal in my opinion, but you absolutely cannot exceed the weight rating of those tires. Your rear axle is actually the same one they put in a SRW 3500, so I wouldn't worry about going 1k over that, but you will have to replace the tires, and possibly wheels (check their rating).
You can carry a lot more than the raw GVWR numbers would suggest, but you are going to need to look at some fivers a little smaller than you hoped for. There are plenty of good models that will come in at around 12k GVWR, and that's realistically about your limit on that truck. And only assuming you are okay with exceeding the truck GVWR, which is a choice only you can make.