Is a weight distribution hitch needed on dually?

You can proof the tongue load numbers, but I wouldn't have an issue with hook and go with that trailer with no sway or wd. Unless there's something different and the trailer has abnormally huge tongue weight.

There are 2 schools of thought. 1 is the typical RVer school where ALL TTs need a wdh with sway control.
2 is the rest of the world that haul trailers of all shapes and sizes day in day out without much concern, especially with reasonable and moderate loads like your camper.