TLD: Scaring (and hunting) a cougar
At least idle cougar can be scared with a flare gun (didn't try it on a charging cougar and don't want to try it). That can help if you want to stealth hunt it.
But it's blind shooting so you should save the game before (i.e. pass some time) and prepare to spend some time reloading and trying again and again.
I did ~ 10 attempts and 3x I successfully scared the cougar to run.