While I didn't find many extra bullets at all (outside of Mystery Lake), all the extra clothing I found meant that I could track the bear all day without freezing. The only place I didn't go was Timberwolf Mountain, but I find it hard to believe the challenge would require this much travel.

I've tried starting the challenge by trekking all the way out to Hibernia Processing and searching everywhere in between for extra bullets. I'm guessing head shots are probably not really effective here? After 21 shots of this type the bear is still alive and I'm out of bullets. So then I tried hitting every shot to the bear's side while he is unaware of me. So I figure maybe he is much less vulnerable to bullets when he is in attack mode. Shot him 20 times this way over the course of the challenge and ran out of bullets. One thing I tried was having every shot in the middle of the bear's chest as he is rearing up to attack.

I think there are probably certain conditions where your rifle shot will do maximum damage to the bear, and I've been trying through trial and error to see what they are.

Yeah, I've been doing all kinds of elaborate things trying to complete this challenge and nothing seems to be working.

The long dark bear challenge map