We just did 3000+ miles to Canada with 6 ADULTS in a 19' Rockwood Roo towed by a Suburban 1500 with a standard trailering package, but w/ aftermarket tranny cooler. I bought the HTT for the same reasons you specify: Great weight/bed ratio, a refrigerator, and my own toilet.
It takes some thought and careful planning, but is doable. However, since your family is not gonna get any smaller and lighter, I would be looking for a 2500 Suburban. The only 2 major downsides to my rig:
-Gotta carefully monitor weight if you're going "full load". Read up and be smart on this topic. It is critical. Your Suburban can pull a more than it can stop, and may you may cause damage to your vehicle or trailer. (I was 700# over GCWR, but inside limits on all axels/tires, and carefully balanced on the scale.)
-You don't have much storage. You've got the 3rd row seat filled with people, and the trailer doesn't hold much. Pack accordingly. Bicycles are pretty much out of the question, for instance.
Those downside aside: With the equalizer hitch, she tows like a dream and can pull even the biggest of mountains if I take it easy and use the truck lane.
Of course, once I'm free of a couple of teenagers, I see a 2/3500 pick'em up with a 5th wheel in my future...