My last 1500 pickup had the 5.3L, 3.73 axle, only a 5 spd and an aux. 41 gal. fuel tank. My trailer had a GVWR of 6550 lbs and I had it loaded within a few hundred lbs of GVWR. So, I was around 12K combined. While overall the combo did fine, there was no power margin for adverse conditions. Couldn't run the A/C while climbing a moderate grade and with strong head winds, couldn't maintain speed with the A/C on. This resulted in more frequent shifting and driver fatigue. OK for a few hours, but long trips get really, really long, and not just for the driver plus greater difficulties at the end of the day setting up camp.
I tend to agree with others, a more capable tow vehicle and a larger TT will make the entire journey more pleasant and rewarding. People suggest keeping weights under 80% of max. for many reasons. Plus with full-time travel and a large family, you will add weight as the trip progresses.