Your body is sedentary when asleep, and brain function and biochemistry change- that is what makes the heart rate drop.
'Battle breathing' as you call it, is a relaxation technique designed to minimise stress hormones like adrenaline that amp your system for fight or flight. Again, dealing with this has nothing to do with the amount of oxygen you take in.
There is no such thing as 'belly breathing'. Breathing is the intake of air in to the lungs, where gaseous exchage occurs via the alveoli. The lungs are actually massively over specified for our requirements, given that even under the exertion of physical stress, we do not use 1/3rd of their intake capacity.
Believe what you want, but belly breathing does not give you a significantly higher volume of oxygenated blood.