Sounds like you'll be at or over your limits. 6 humans, plus gear, hitch. You'll likely be around 8500# ready to camp on the trailer. Do your math and remember kids have a funny way of growing,
As far as "no other TV options" we camped with a family of 6. They pulled with a CC picked with front bench. They even crossed the country that way so it can be done. A properly optioned 3/4 ton gas pickup would offer both more towing capacity, more payload and a longer wheelbase. All those thing will make for a better towing (safer) experience. Not what your original question was but sharing another family's experience that is the same size as yours.