The hard point of the car is part of the frame on every vehicle I can recall. And I recall a lot, because back when I was a mechanic I had to find those points to get the car on the hydraulic lift. One had to be careful of rusty VW Beetles as unibody as kind of a new thing, and a lift could punch a hole through the floor of a rusted Beetle if one wasn’t careful.
Anyway, enough resume flouting. Yes, you should unload your truck before changing the tire. What if that 1000 lbs. is pea gravel? Still unloading it first, Dr. Safety First? If I had 1000 lbs. in the bed and I had a flat? I’d change it as is. It won’t be the smartest thing I do that week, and I am going to keep my hands out from breaking the vertical plane of the wheel well as much as possible. But I am confident that the solid point of the frame will hold it, and even the puny jack is likely stringer than it looks.
But in reality, if I had a load, I’d just call AAA or a mobile truck service. They carry really nice heavy-duty jacks. Flat tires are such a rarity these days, I’d roll the dice on paying someone $200 to change my tire.
In the end, if you’re that worried about it, do what we did when I was kid with our old ‘63 Chevy: throw the spare on top of the load when you’re done loading. Doesn’t fix the jacking problem, but at least the spare is accessible.