My King Ranch is even more challenged than your truck. My GVWR is 7200, and the sticker payload capacity is only 1115 lbs - but with my drop-in bed liner and BakFlip tonneau I'm down to 1020 lbs!!!
With the hitch weight of my 6200 lb car hauler at about 675, and my almost 100 lb Equal-i-zer WDH, when I put my 220 lbs in the driver's seat I've got maybe another 25 lbs before I'm overweight. Actually, when I tow 700 miles up to a track event later this month I'll probably haul a different car that will add about 250 lbs - so my loaded trailer will be almost 6500 lbs.
However....another limiting factor in pulling a trailer is the GCWR. The GCWR for my truck is 15,500 lbs.
if I load my truck to its 7200 lb GVWR, then subtract that from the GCWR, I can pull 8300 lbs.
Butt....how can I do that with my very limited payload capacity???
I got a Trailer Toad!! The used Toad I got has a 3500 lb axle, and can only handle about 2000 lbs of tongue weight, but I won't get close to that. The new Toads can handle more than 4000 lbs of tongue weight.
With the Trailer Toad carrying the entire tongue weight of the trailer, I have the entire payload capacity of the truck to haul me, my wife, her luggage and snack cooler in the cab, and I can still load up more that 500 lbs of tools, spares, and track support junk in the bed of the truck - but I'll be hauling less than that so we won't even be up to the truck's GVWR.
If I was at the truck's GVWR, I'd be limited to towing 8300 lbs. My Trailer Toad is about 200 lbs, so I could still pull a trailer that weighs more than 8000 lbs and not exceed any limits of the truck.
That's just my solution to the towing problem of my truck having a VERY
low payload capacity.