While I was shopping for a new truck to haul my 6500 lb TT, I noted that a brand new Ford F-250 Crew Cab with diesel engine only had a payload rating of 1250 lbs! Those diesel engines are heavy and significantly reduce payload. Put 6 people in that truck and you are already at payload. I ended up buying a F-250 with gas engine and payload of 3400 lbs.
As others have pointed out, you are wise to realize that payload is the most common rate limiting factor in many trucks.
I suspect that as your family grows, and you accumulate more gear to stow in the truck and TT, you will be happier with a payload in the 3K or more range.