Forget about the cougar, because you would never see it coming.
Not true in my experience. I've seen it coming, twice.
First time wasn't enough for me to have a "come to Jesus" regarding carrying guns in the back country. It took a second encounter before I dropped money on the notion.
Perhaps I'm in the blessed minority that has a repeated tendency to be stalked by mountain lion, and lucky enough to notice and do something about it.
I'll tell you that I "got by" with building a relatively wild, large, uncontrolled fire and was lucky I didn't burn down the forest both times. If I had a pistol, there would have been a dead cougar each time. And I'm 100% A-OK with dead cougar, when they reveal themselves as potential man stalkers. The first one tripped over one of my tent lines in the middle of the night and I scared it off with a flashlight, then lit a big fire. The second one (two years and 1500 miles away) screamed around the edge of my fire's light and I think it wanted to eat my 5' tall girlfriend at the time. It screamed around our camp for 10 minutes, less than 50 feet away from us around the giant fire.
OP:
If your southern black bear get no bigger than about 300 pounds, then a 9mm with FMJ-FN heavy ammunition will do okay. The goal is to hit the sinus gap in the skull below the eyes, behind the nose, when they charge. Cougar are not nearly as tough as most folks make them out to be. As long as you can get a shot off. They are spooky fast, and it takes good luck and awareness to even realize if you have one eyeing you.
However, I will not carry a 9mm in the back country, even here in AZ. We don't get huge bear here (maybe 400-500 pounds max, though 250 is about the norm). I'm just a believer in "wide meplat, heavy bullet, enough velocity" theories. Elmer Keith's
Sixguns resides on my coffee table. I carry a 9mm around town, but in the woods I carry either a LW Commander in .45acp when weight is a concern and the environment doesn't support large critters, or a .44 magnum Redhawk when I have to have that extra power.
A "Wilderness" belt does wonders for back country support of a pistol.
Add to that an OWB holster with a wide belt mounting system that distributes the weight of the pistol around the belt as much as possible... rather than tiny narrow little loops that sag at your hip.
Another vote here for re-examination of your carry rig and do what needs doing to make the .44 your partner in the back woods and swamps.