Welcome OP!
You’re on the right track and a half ton with a big motor and crew cab will do everything you need with a reasonable size TT. 6-7klbs is not a stretch for any newer half ton that has good power.
You’ve already heard the “you need diesel” comments. And you’ll soon hear the litany of how you can’t haul that much weight, 3 kids and a camper with a half ton due to gvw limits.
Both are not horrible recommendations but both are not totally true either.
Think about your use of the new tow rig outside of that one trip. That needs to be considered as much or more than the trip.
Sure I have a tuned diesel and love the pull anything power. But if budget didn’t allow I’d have just a half ton gasser for most duties. I have one of those as well, have for the last 25 years and I tow more with it than the diesel. (Company vehicle and generally towing for work).
Again, any big half ton set up right is totally sufficient for your goal.