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Hauling about 4100 lb and towing a 5000 lb trailer.
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F-250 is out, so let's start with the F-350 SRW. With CrewCab longbed 4x4 PSD, GVWR is 11,500 and GCWR is 23,000. The wet but empty truck is going to weigh around 7,300. Add another 1,000 pounds for toolbox fill of tools, cooler full of cool, driver and one passenger, and various "stuff". So 8,300 pounds for the wet and loaded truck. That leaves 3200 pounds for payload. Not enough for your needs.
Same truck but F-350 DRW. GVWR is now 13,000 pounds, and the wet and loaded truck weight went up about 400 pounds to 8,700. Additional payload capacity 4,300 pounds. Not quite enough for 4,100 pounds plus about 600 pounds of hitch weight.
So yes, we need an F-450.
GVWR 16,000 pounds and the truck weight went up another 500 pounds to 9,200 after we add a bed or tow body. So 6,800 pounds of additional payload. Plenty for 4,100 pounds on the truck plus 600 pounds of hitch weight. And have 1,100 pounds of payload capacity left over for graduating to a heavier trailer or other payload in the future.
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I wonder about the front axle capacity...
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No problem. 6,000 pounds GAWR on the F-350 DRW. 7,000 pounds on the F-450. That's plenty if you don't exceed the GVWR of the truck.