Your body will react to the lack of air very quickly. Hyperventilating slightly, then exhaling 2/3 of the way, holding your breath at this point until well after the shot is what has worked well for generations of shooters. Most of those shooters did not compete by climbing mountains or running long distances to firing positions. If you must exert yourself to the point of heavy or labored breathing prior to the shot, then either take long enough to return to normal breathing, or develop techniques to either mitigate or isolate your breathing from your rifle.
Hyperventilating is a reasonable solution for light exertion, but won't work for heavy exertion combined with a shot demanding rapid execution. Learning to use the 'Ambush Trigger Method' explained elsewhere in this document, will allow you to make remarkably quick and accurate shots under stressful conditions.
The 'Free Recoil' technique will allow you to totally isolate your breathing from your aiming. To accomplish this, no part of the body except the trigger finger (and maybe the thumb) is in contact with the rifle. The body can be in the conventional position, with quite small gaps where normally contact would be made. 'Catching' the rifle with your shoulder with heavy recoiling light weight rifles requires a lot of training before relying on it in a match or in the field. Alternatively the body can be totally re-arranged so that no conntact is possible, and recoil is dissipated through a lot of space behind the rifle, and friction from the supporting system.