You can tow the 6200 pounds provided you do not exceed the Gross vehicle weight rating The figures are all on a sticker on the drivers side door jamb.
It may give you your payload capacity directly or you may have to subtract the curb weight from the GVWR.
In the 6200 lb trailer example between 10 - 15% of the weight of the trailer is transferred to the vehicle so about 900 lbs (sometimes called hitch weight or tongue weight) the hitch itself will add about 50 lbs to the vehicle. You have used 950 lbs of your 1600 lb capacity. now everything you put in the trailer will add to the weight. So if you add 1000 lbs of gear to the trailer then about 100 lbs will transfer to the vehicle. A full propane tank weighs about 40 lbs and because it is so far forward all of it will transfer to the vehicle. Battery 20 lbs transferred to vehicle. So minus another 150 lbs. This means you have about 500 lbs of capacity left. With your family you are probably just over the limit. So nothing dire will happen if you are over but your towing experience may be unpleasant and handling may suffer. If your intent is to just go locally for weekends and such is one thing, but I would not be going cross country