@Nex I think is on to part of your issue. I can’t see the load rating, but those are KO2’s. If they’re the stock C load range, the psi makes a big difference in compliance - and economy but that’s a secondary concern to traction. Check for the “load range” on the tire sidewall, make sure you’re campaigning Load range C tires. The E range tires will work, but the ride will not be as compliant.
The door placard and manual say 38 x 38 psi. Raptor Assault recommend I think 32 x 28 psi Front to rear for mixed use off roading with some street travel. I like about 35 x 33 or 32 front to rear for street biased driving with some off road. Any higher and I get a rougher ride, more suspension impact from smaller obstacles and more disruption to forward motion over broken terrain. I don’t like over 36F and 34R cold because of uneven wear; the middle of the tire will wear a lot faster. And if I’m going to be spending more time off road, I’ll drop psi further - 33 / 30 in and around the beach and I’ll drop it for sand use.
The ‘construction sites’ use case concerns me. Yes, I know that a lot of the construction sites will permit use of even street vehicles but there are also sites that are ‘unimproved’ and can be quite rough. I’d be a bit concerned with less than zero offset.
Our supporting vendors here are great for getting our trucks set up with new shoes. give one of them a shout and see what shakes out.