I think you'll have a hard time making any of the current full size SUVs work. That's not a super heavy trailer but it isn't exactly light, and it's very long. If it were me I'd be looking to pull it with a heavy duty truck, especially with the family on board.
We have three kids and the truck in my sig fits us great. A megacab would be even roomier if that interests you, although you obviously even with that you won't get a third row of seating. We have a fifth wheel so an open bed is a requirement when we're towing, but I have a Softopper for trips without the trailer and it does a great job of instantly giving us a huge protected cargo space for our stuff and the dog kennel. Since you have a travel trailer you could get a standard fiberglass topper that would give you a little more security.
I'm of the belief that pickups are a much better value than full size SUVs unless you really need that extra row of seating, with the big bonus being that you aren't restricted to a light duty vehicle.