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Your SOL on tire size the rim is too narrow for any other tire than what you have. You can try putting 255/85R16s but you will have good sidewall bulge as the rims are only 6" wide.
The 10 bolt wheels that the F-Superduty (F-450) trucks have are only in that size so you have no other options.
What do you want to put larger tires on the truck for anyway unless your trying to make the gear ratio a little taller. I already did the number calculations and it doesn't make much difference when you have 5:13 axle gears.
You can go with Rickson 19.5 wheels they have them for the 10 bolt pattern used on the F-Superduty trucks they are not cheap. I looked at them to get rid of the 16" tires the 19.5 tires wear better.
If you want a truck that will carry the payload then you will need a wheel that will have the same 3000lb weight rating. The old F-Superduty trucks can carry 8000lb payload almost legally had more on my trucks but don't do that too often you run out of braking power.
The F-Superduty trucks have decent brakes but load them up to 17,000lb gvw then your pushing your luck. Even at 15,000lbs which is legal the brakes are a lacking a little.
I am just looking for a little bit taller tire. Dave, I like the link you posted. A little expensive but it sure would be nice. The 19.5 conversion waould be nice also but I can't really afford to lose any of my towing or payload capacity.
You wouldn't loose any payload capacity with the 19.5s they will actually exceed the capacity of the truck. The benefit of 19.5 rubber is it lasts longer and the truck handles a little better.
The only thing with the 19.5 wheels and tires is your wallet will be a little lighter
I am running 255/85/16 on my F350 and first off it looks really cool with the chrome simulators, but you need just slight spacers, the kind you buy from a trailer axle supply place, not the big ones like you need for 285s, just little ones, I think they are 1/4 inch, but with too much of a load they still rub, so I'm looking for what I need to go slightly more space without going way out of budget, cause the budget is already stretched awfully thin!
One of my friends here has a 2002 F350 dually welding rig.
He goes out to well locations and pipelines.
Got stuck everywhere and had to be pulled by a dozer.
He put a set of spacers like you would run on a single rear wheel to run duals on it between the wheels.
Added air lockers to both axles.
Added on board air and Ride Rite air bags to both ends.
He is running 315/75-16 mud tires and has all kinds of room between the tires.
Now when he leaves the paved road he cranks the air bags up, hits the lockers and goes.
No more getting stuck in the mud.