The LC-580 is a 10.6ft truck camper..I have the LC-480 tagged by lance as a 2660# camper and weighed appears to be spot on..My point is my DRW F-350 has 4600# of manufacturer rated payload and I am within a couple hundred pounds of being maxed out on GVWR on an F-350 longbed ready to camp...6K on the rear axle for my 9.6 on a longbed,you would be more for my TC on your truck because it is a shortbed..The 10.6..More than that.
If you had a longbed,I would say go for it as Specta does the same with a 11.3 TC on a 3/4 Chevy longbed in the same erra as yours.. I have seen 3500 series(1-ton) trucks with longbed Truck Campers(8.6ft) on shortbed trucks without any issues but never a TC as long as that one(10.6) on a shortbed...
Might wanna load it up and see how it fits and take it for a short drive empty to see how it handles first..That's what I would do on that TC.