With my signature trailer weighing approx 6k before any toys, I was on the verge of overloaded with a '97 K1500 (350/4L60E/3.73). GCW was 11600lbs. I had added a B&M cooler IN ADDITION TO the factory aux cooler along with a trans temp gauge that I located in the pressure test port.
It was hard to keep the torque convertor locked in 3rd. At 65mph I was a little bit under the peak torque of the engine. If the 6000-6500lb load was on a flat bed and didn't grab much wind it would have done ok, but being an enclosed trailer was just too much work. Engine power output was adequate but transmission is not there. I even toyed with using a manual TC lockup switch just to keep temperatures down. 4.10 gears would have helped but not worth the money to swap out. If my truck would have been the NV4500 5spd man trans I might have kept it a little longer, maybe.
Anything other than flat ground and I had one eye glued to the trans temperature. If you plan to go farther than 50mi your transmission is going to hate you and will prove to you the hatred it has in a short amount of time.
I bought a larger truck 2 months after buying the trailer. Same drivetrain as you and I have less trailer.
2005 2500 Cummins/48RE/3.73, QCLB, 4wd, BigHorn, Edge Juice w/ CTS + Turbo Timer,Transgo Shift Kit ISSPro Oil and LP pressure gauges, GDP 20/2 filters, Custom Diesel Steering Box Brace
'10 Forest River Shockwave Toy Hauler 21'
Honda EU3000I Genny