The 6,572 dry weight of the camper will be increase significantly after you add battery, propane, weight distribution hitch and all the stuff you’ll need for a family of four in the camper. You also need to pay attention to the maximum payload for the truck because in a lot of case that becomes the limiting factor. If you end up with a real trailer weight of 7,500 lbs 10% to 15% of that weight will be carried by the truck so you may be looking at 1,000 lbs (tongue weight) plus the weight of four individuals and whatever else you carry in the truck as your final payload number.
If you can add all of those things together and not exceed the maximum tow or maximum payload numbers the truck will manage the pull but performance (acceleration, climbing steep grades while maintaining speed and stopping) will be marginal.