The problem is that you probably will have 300 - 400 pounds of passengers, and 10% of the 9,000 cargo trailer is around 900 pounds on the hitch. So that is 1,300 pounds taken away from the 2,300 pound limit. It only leaves 1,000 pounds of the camper.
Some campers are in the 3,000 pound range, so a dually will solve all those problems, even if the 9,000 pound trailer is hitch heavy and 1,400 pounds or more on the hitch.
Sadly, you do need a larger capacity truck. The SRW F-350 might take care of the problem, however depending on the camper that you pick (a Host with dual slide outs in in the 4,200 pound 'wet' weight range) you would have to work around the cargo weight limits with any single rear wheel truck. It is only a dually when you can safely say that the truck can carry just about any load.
Good luck,
Fred.