With a family of four and that trailer you should have no issues finding something that's comfortable and has the towing capacity.
That trailer will be in the 6500-7000lb range loaded and ready to camp. To be comfortable on the road you really need something rated to tow much more than that.
It's been my experience that a manufacturer's tow rating should include the footnote, "but you don't want to go very far towing this much." The more excess capacity you have the better it's going to feel, and you want to LIKE driving this rig or else it's a total waste of money.
I'd personally go with a truck rated to tow 8500lbs minimum. Lots of Tundras and F150s and 1500s are capable of that.
Just be careful when shopping and make sure that the tow rating is for THAT SPECIFIC truck, not an advertising "up to" number on a piece of literature. Tow ratings will be in the trucks' owners manuals, and will be based on engine, option packages, and axle ratios.
Look for features such as 6-speed transmissions with 3.42 or 3.73 axle ratios, engines in the 5.X Liter range, payloads over 1500lbs. If you're looking at older trucks with 4-speeds, then look for 3.73 or 4.10 axle ratios.