Although I cannot speak for your vehicle....
I wrote earlier about this subject stating, from the factory, our new PU did not come with the fused charging wire plugged in at the fuse box. Even tho' the PU is trailer equipped, the wire at that location wasn't connected, and actually isolated. I'm guessing that you may have this same problem.
You should consider reading the earlier post regarding relays, should you decide to install, or have one put in. Some may ruin your computer/DIC in the PU, so check with an authorized dealer before you have your RV people, or you, stick one in.
Otherwise, I'd certainly do as previously suggested, use your volt-ohm meter at your PU and trace the problem this way.