Issue per say is NOT the GCWR or tow rating, but IF you have the chassis and payload for the load you are going to tow! Frankly, a family of 6 will usually have most 15 series rigs at gvwr before loading a trailer on board. You will need as "I" needed, to be one or two gvwr sizes up from a 15 or half ton style truck/suv etc.
A 2500 burb with the 6.0 gas, 6 sp trans will tow your current trailer, along with carry your family and be very safe! and reasonably comfortable. Reality is, the MOST room and comfort will be in a van. BUT, if you want 4wd, unless you are willing to try and find on of the few conversions out there in the used market, or pay a premium to have an 8 lug van converted, right now the only option in an SUV/van is the 2500 burb. OR< as i did, I went with a crew cab pickup for my family of 6! With no regrets mind you.
I was typically 4K over gcwr, but under my axel wt ratings, GVWR I bounced from 200 under to 200 over. That never worried me either!
Marty