The smallest Jayco toy hauler has built-in fuel tank and generator prep, but it will severely overload your F-250.
Floorplans - Seismic Toy Haulers - Jayco
The Eagle HT you are considering has a GVWR of 9,950, which is about the max you can expect to tow with your F-250 without being overloaded over the GVWR of your truck.
There are lots of ways to skin a cat, so here is one way for you to proceed:
1. Order the Eagle HT floorplan you like best. They all have GVWR of 9,950, so no difference in hitch weight of the different floorplans.
Floorplans - Eagle HT Fifth Wheels - Jayco
2. Order a Honda EU3000I inverter generator. That will be enough power to run one AC during even the hotest day.
Honda EU3000i HANDI Generators Online WiseSales.com
Not cheap, but much, much, more reliable, durable, and quieter that the cheaper generators. Been there, done that.
3. Plan on having a local welder (or maybe your RV dealer) add a class III 2" receiver hitch on the back of your trailer. The HT is a very lightweight 5er, so the welder may have to add a lot of junk iron under the rear of the trailer to add enough frame/chassis strength to support a class III receiver rated for 500 pounds of hitch weight.
4. Order a luggage/cargo carrier with 500-pound weight capacity that plugs into the receiver hitch.
24x60 Railed Cargo Carrier for 2" Trailer Hitches Pro Series Hitch Cargo Carrier 63153
5. Order two or three 5 gallon metal "Jerry" cans for safely hauling gasoline.
Amazon.com: Midwest 5 Gallon Metal "Jerry" Gas Can: Everything Else
6. I also carried a 30' 30-amp extension cord, so I could move the generator further away from the RV during snooze time.
As soon as the trailer arrives, have the receiver added to the tail of your 5er.
The EU3000i will weigh around 110 pounds full of gas and oil. Each 5-gallon Jerry can full of gas will weigh about 50 pounds. So the generator and 3 cans of gas will gross about 260 pounds, or much less than the 500-pound capacity of your cargo carrier.
That extra weight on the tail end of you 5er will slightly reduce the pin weight of your rig, so that should help if you tend to be crowding the GVWR on your F-250.
Be sure you secure the generator and gas cans to the cargo carrier so they cannot get loose and come off during the trip. We used a heavy chain and padlock to be sure someone didn't "borrow" them, and to keep them from bouncing off the carrier.
We did something similar to the above suggestion for several years of dry camping at Texas Motor Speedway during race weekends. The only difference was our Honda generator was the more powerful but louder and much heavier and thirstier EM5000S.