The smell of breathing in dead animals should not cause you to get sick.
Although breathing in a dead animal smell may make you temporarily sick to your stomach if you have a weak stomach and you may vomit or feel like vomiting.
But other than that as long as you don't hover over the dead animal for hours breathing in the dead animal smell there should be no health risk.
I've breathed in dead animal smells before when I used to pick up dead animals off the road as a job.
But never have I gotten ill from breathing in the dead animal smell and I never wore a mask even though I was supposed too.
Mostly the dead animal smell is just unpleasant but not harmful to us humans.