Care to explain why fight or flight primes the body to take in more oxygen by increasing breathing rate, yet this increase in gaseous exchange fuels, not quells adrenal dump?
Deep breathing is a relaxation technique, and no matter how you engage the diaphragm, the act of respiration comes from the lungs, not the 'belly'.
Elevated breathing rate is the one aspect of stress physiology that you can control. By doing so, you send conscious signals to the unconscious physiology that it can relax. The adrnaline dissipates, and that is why pulse rate drops. It has nothing to do with oxygen saturation levels.