Each time I've shot a deer, there was much more blood coming from the bullet hole than the face. The lungs don't have to fill up to bleed from the bullet holes. The bullet hole is where the damage has been done. They are going to bleed from that wound. Your blowing mist theory has some "holes", too
. When the deer tries to breath, air is escaping out the entrance and exit holes. Especially if you have a big exit wound.