I don't think this works. To see why, suppose we make the tires out of a very stiff material. We control the pressure P, but changing the pressure will not change the surface area A. Therefore, by changing P we can set F to whatever we want, which shows that we are not actually measuring the weight of the car. The basic issue here is that the air pressure is not the only thing that is supporting the car.
I don't know how important the stiffness effect is for real tires, but I suspect they are sufficiently stiff that it matters a lot.
For the case you suggested above, this is essentially how run-flat tires work. Of course, the sidewall of the tire isn't as stiff as the support ring, but they do provide some support.
Let P_m be the pressure recommended by the manufacturer. A better sanity test would be to fill the tires to .75P_m, then to 1.5P_m, and see if the area in contact with the ground has shrunk in half.
It's odd, because as I was reading your comment, I was thinking the exact opposite. I am considering the force it takes to squish a flat tire. A person can squish a (regular car's) flat tire with his/her hands.
My gut-check tells me that the stiffness of each tire can lift less than 10kg/20lbs... multiplied by 4, that still is not a huge contributor to the overall weight.