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.
2011 Cougar 322QBS
2007 Pilgrim 278BHSS (Sold)
2023 F359 CCLB 7.3
2013 F350 CC LB 6.7 (Retired)
2002 F350 CC LB PS (Retired)
B&W Companion Hitch