The "elevated heart rate" status is tracked at the time the animal falls. Meaning you DO NOT have to have a high heart rate when taking the shot which downs the animal
Once you know how this works this achievement comes easier.
I personally unlocked this achievement by scoring a vital (lungs) hit on a Moose with the starter rifle.
It will run for a while, but as long as you sprint after it, your heart rate should be high when he finally falls to bag you the achievement.