There is really nothing that can replace weighing the actual truck, and subtract that weight from the GVWR. The resulting number is your official payload capacity, no matter what is listed anywhere*. But even then, the payload capacity is only one number among others and needs to be understood. It is well possible to overload the rear axle and remain within the overall GVWR.
The axle weights are actually the more interesting numbers. Typically, the sum of the weight ratings of both axles is a number which is greater than the GVWR. You can make the point that it is in some way OK to exceed the GVWR as long as both axles are within their weight limits. In fact, the german authorities (thats where I live) will in many cases accept that calculation and put it into your vehicle's paperwork. But that is another story.
Payload is a good indicator how much camper your truck can carry. But it is where you carry the load which is important. Imagine a crew cab, short bed truck, with an 8 or 9 ft camper in the back, and possibly a trailer with a hitch extension behind. It is easy to see that much/most of the added weight is behind the rear axle. The further aft the center of gravity of the load is, the more of the added weight goes to the rear axle. If the center of gravity of the load is behind the rear axle, it will actually take weight OFF the front axle... So, in this case, ALL of the added weight, plus the weight removed from the front, goes to the rear - after all, it has to go somewhere, as the truck didn't get any lighter.
Nothing beats knowing the numbers for your particular truck, and weighing the actual loads at a scale.
* provided the truck is in stock configuration (wheels, tires, springs etc.)