Tire weight carrying rating is based on the maximum tire pressure that is marked on the sidewall of the tire. Running tires underinflated isn't good for the tire, but it might make the tractor ride a bit smoother. An example that comes to mind is the late 1990 Ford Explorers, where Ford recommended a lower tire pressure to give a smoother ride. It also led to tire failures, along with a recall of the tires themselves. I always run tires at the maximum recommended tire pressure, and at that time, when Ford replaced the tires as a result of the recall, I had over 40K miles on my tires, and no failure.
I have a BX22, and my rear tires are filled with liquid ballast. Since my tires are about 15 years old, they loose air, and I can immediately tell, because the rim (wheel) will spin inside of the tire, and I will not have any traction. I air them up to 20 PSI, and the traction is restored. It is a tractor, and it was never intended to have a smooth ride. The ride is determined by the ground underneath the tires, because tractors don't have any suspension components. Everything is bolted solid. If the front tires aren't inflated to the maximum, when you are carrying weight in the front bucket, you run the risk of the front tire coming off the rim. If this happens, and you don't notice it almost immediately, you have a good chance of destroying the tire beyond future use.