The air in the bladder will be at exactly the same pressure as the water around it. Therefore if the diver were underwater with the bladder (a thing which I do

The air in the bladder will be at exactly the same pressure as the water around it. Therefore if the diver were underwater with the bladder (a thing which I doubt, for reasons others have mentioned here), the air in the bladder would be at about the same pressure as the diver's lungs (about 0.2 psi higher, if the diver somehow managed to keep the bladder below him, as in the picture).