I have another take on this. How open are you to "thinking outside the box"? The travel trailer box that is. A very close friend had a family of 7 and never owned anything bigger than a 14' pop-up (which no one makes anymore). The oldest kids slept in tents. Pretty much all his kids still love camping as adults.
I'm not suggesting a pop-up. I suggesting you might want to consider putting the older kids in tents. For payload reasons you'll still need a 2500 series 'Burb or equivalent, but using tents for some of the family members would allow you to consider smaller trailers, perhaps a hybrid trailer, and have more margin for payload on the truck. Not to mention that the kids will love sleeping in tents.