Without knowing all the truck and trailer weight capacities I would still say it is likely too much trailer for your truck, especially if you plan on carrying that large family in the truck too.
A 7500lbs dry trailer will push 9k once loaded with a tongue weight pushing 1200lbs.
Doesn’t have to be a diesel but I’d bet you’d need the payload and ratings of a 3/4 ton truck.
I have the same layout in our Bullet 31BHPR but it’s delivered weight is only 6400lbs. We towed it for 2 years with a more capable 1/2 ton than you list and it did ok but we were overweight even when lightly loaded. We now have a 3/4 ton and it’s a much more enjoyable experience. I would never recommend our size trailer with a 1/2 ton.