The tow rating of your truck (9,600 pounds you say) is based on an empty vehicle with a 150 pound driver. As you add the weight of people, truck accessories, and cargo the rating technically goes down.
The problem is the payload capacity of your truck is rather low and TT tend to have rather high tongue weight. A boat on a trailer or a flatbed hauling equipment, for example, will have a lower percentage of tongue weight than a TT. You might be able to tow a 9,600 pound boat but not a 9,600 pound TT. This is something the vehicle manufacture usually doesn't mention or hides in the fine print.
I don't know the axle weights of your truck, the payload of your truck, or the tongue weight of your fully loaded trailer. But, my guess is that when you are ready to go camping you will be exceeding the payload and likely the rear axle weight rating of your truck.
