For your calculations you need to:
1. Subtract the weight of any aftermarket accessories, occupants, pets, and cargo that will be going along, in or on the vehicle, from both your payload and tow rating numbers. Subtract another 100 lbs, if you're going to use a weight distributing hitch. The remainder of both payload and tow rating, are your max capacity for trailer tongue weight and towing weight.
2. Average camping load of dishes, pots and pans, bedding, camp chairs BBQ, tools, groceries, and water, etc, is 800 - 1000 lbs. Add this to the trailers unloaded weight.
3. Average loaded tongue weight is 12 - 13 percent of loaded trailer weight.
The trailer you are looking at, will have a loaded weight around 5000 lbs, with tongue weight in the area of 625 lbs. That would put you in the weight range requiring a weight distribution hitch.
The manufacturer's tow rating, allows for weight of a driver only. Any weight added by accessories, passengers, or cargo, reduce that rating, pound for pound. Most likely, you would be looking at towing a 5000+ lbs trailer with much less than 5000 lb tow capacity, and possibly over weight on payload.