"When you would gain any amount of fatigue, you may instead place 1 HP chip on this [pet]."
For lockpicking, adjusting a single die is considered an instance of gaining fatigue. So you could adjust a 1 to a 6, and instead of taking five light fatigue, you could put a single chip on the Palefrost Elk. However, if you need to adjust a 3 to a 4 and a 5 to a 6, you are adjusting two dice so that is two instances of light fatigue, and you would need to put two chips on the Palefrost Elk to avoid both light fatigue.