Generally unnecessary.
Every time you water your soil, it draws-down air via suction. If the soil is too muddy, and won't allow for a good exchange of oxygen to the roots, perhaps use more perlite next time, or water less often.
H2O2 is a good way to sterilize the soil. It can easily kill all the beneficial bacteria if overused.

The simple act of stirring (agitating) the nutes into the water, when just mixed...is fine. There is only so much dissolved oxygen (really tiny O2 bubbles) you can create/capture to add to the soil anyway. Most is lost as soon as you shut-off the pump, and add the water to the pot. (water wants to go down, O2 wants to rise)