A gut shot deer can be just as edible as any other deer. You are the victim of an old wives tale.
You want to field dress the deer, and then wash out the inside of the carcass to get rid of stomach acids and enzymes that may have gotten into the cavity because of piercing the small or large intestine, or bladder. Then dry the carcass with towel, or paper towels. Cool and remove the skin. You actually should be more concerned by bacteria introduced to the meat from the projectile pierceing the skin, and taking dirt, fur, and hide into the meat.
All muscles( meat) are surrounded by thin, whitish tissues that are either tendons, or ligaments, or just connective tissue. Mother Nature covers the muscles so that any infection in one muscle- due to injury or disease, cannot spread to the next muscle. You don't have to worry about the spread of stomach contents from a bowel shot, but you do need to worry about the effect of the stomach acids, and enzymes from the intestine( bowel) on those connective tissues. Once the muscles are exposed, they can be infected.