Ok, take a deep breath and figure this out.
I'm normally the guy that says "you can tow that, done it many times", but a decent size lightweight TT to confortably sleep 7 is, like you suspect, going to be 7-8klb min, 1000lb tongue weight and your vehicle cargo is getting bigger every year.
Soft suspension, Navigator built for cruising not heavy towing. And you'll be overweight on all accounts before the last 3 kids get strapped in and the water tanks filled!
Wrong vehicle on several fronts.
I got 2 kids and can fill the back of a pickup to go tent camping for 2 days, betting you don't travel light either, lol.
Distractions while driving are numerous and unexpected sometimes w 2 kids. You have 2.5x the distractions combined with presumable inexperience towing, nevermind in the mtns too. You want a rig that will handle the camper, not the other way around.
Horsepower. You're down a few ponies sitting in your driveway, but by the time you make camp at 9kft after climbing those passes to get there, you'll wish your tow rig had a tow rig. That 5.4 triton would have a long day doing that duty eastbound across Kansas with a tail wind. Plus trans coolers, hard on light duty brakes coming off the mountain, etc.
SUV wise, only 2 I'd even consider. Diesel excursion or a 8.1l big block suburban and the sub will suck a little wind at altitude. Plan on some mild suspension mods with these too, but they have the chassis to handle it at least.
I'd keep my eyes peeled for a 10-12 passenger Chevy Duramax 3500 van. Newer ones have a 6 speed trans too. Prolly hard to find the right one, but they exist and are considerably cheaper than the pickup truck counterparts.
Other options.
Get a big arse pop up camper. Half the weight, no wind resistance and will sleep everyone in relative comfort until the kids get big. Can stack them like cord wood on an end bunk!
Get a Class C camper. Not cheap for a big one.
Buy a 1 ton crew cab pickup and make the kids draw straws to see which 2 have to stay home and watch the dog!
Good luck with the search. Think 3/4ton SUV minimum or 1 ton van if you want to pull a decent size TT.
