Toyota tundra?

You need more payload. F150HD is the lightest that could carry your family and the trailer's tongue weight. Or as mentioned a 3/4-ton.

Hi
In my opinion, any TT over 7000 GVWR is better served with a 3/4 ton truck. Aside from tongue weight, the length and sway becomes problematic with softer suspensions and tires normally equipped on 1/2 ton trucks.
While most 1/2 ton truck manufacturers post high tow ratings, the limiting factor is payload. A 10,000 boat/trailer is one thing (lighter tongue weight), but a TT or 5er is a whole different animal. You will notice that most ads show a half ton towing a huge boat, but not an rv.
Steve

Tundra owner here. Is the 7800 the dry weight, or the gross weight? If it's the dry weight, you'll be around 9000 lbs loaded ready to go. In that case, stick with a 3/4 ton, 2009 or later with the 6 speed trans from the big 3. All good trucks.

If that's the gross weight, the Tundra will be OK for the task. If you go with a Tundra, you want the second generation truck with the 5.7 V8 and 6 speed trans with towing package.

You would be right at the limits on tongue weight. Tundra's have a max trailer limit of 10100 lbs some less. You have a 1/2 ton truck which includes tongue weight, all other thrown in truck, riders, etc. Look for something shorter and lighter.

Sadly, you will need more truck.

I own a Tundra, LOVE it, tow a 5,900 dry 25-footer, great combo.
Having said that, I think most replies will address the payload limitation of the Tundra or any other 1/2 ton for that size trailer. The Tundra has the towing capacity, but likely not the payload that you'll need.

not well