You'll need to evaluate what your comfortable with, a lot of rigs are over the limit specified by the manufacturer.
My truck listed the rear axle at 6200# with the tires as the limiting factor. With mods, 3750# tires, I'm running at ~6800#. It's very comfortable for me. Some on the forum would never exceed the sticker limit while other might put 4000# on a F150 with 9 year old tires. (a bit scary)
Some will be fanatical about it and ask you to stay out of "their" state, while others will be helpful and give you advise based on there experience. Evaluate all and remember they're opinions.
As for your truck, my next one will probable be a 350 SRW CC LB 6.2 gas.
I think even if you put a 2000# TC yours you'll feel the sway. It will handle differently with a TC on. So, you may want to re-consider adding at least a sway bar. That helped mine the most but I did't have to have it. You already have heavier springs. If you don't like the feel after you get your TC you can add it later. Or you may decide a DRW is what you want.
I talked to several individual on the forum here before I bought mine who had run OEM size tires with significant overloads for years with no issues. I wasn't comfortable with that so I changed to a heavier tire for peace of mind. Just put a set of Firestones on 2 week ago which ran $901 total.
Again, you'll have to evaluate what your comfortable with.
Good luck,