I've managed to live 60 some years hiking the woods and never been bothered by black bears or lions.
I've seen two grizzly bear in the wild and the solution, spotting them several miles away across a mountain valley, was to high tail it out of there. Grizzly and polar bears are apex predators. I would not feel adequately armed with just one center-fire rifle. I'd prefer my party had three.
Despite bravely running away from those big bears, I feel privileged to have seen them. I'd feel the same about a cougar.
There is a comical youtube video showing someone's encounter with a cougar and his defense with a handgun. The big cat was playing, bluffing charges. But you don't mess around with a big cat to see if it's bluffing.
While backpedaling, the man put a round into the dirt, raising a cloud of dust. He quickly shot a few of the weeds around the cat with several shots. This was not a man giving warning shots; this was a guy freaking out thinking he was going to be mauled.
The cat didn't seem particularly impressed with the noise but loped off anyhow, looking bored.
If that cat wanted the guy, he would have had him. It doesn't matter how big your bullet is if it doesn't hit the target, and the target is obscured in the fog of panic. Anyone that says "Yes, that man was a fool, the trick is to simply not panic" is missing some marbles.
My advice is:
1. Learn your habitat and the habits of the animals around you so you can be respectful and smart about them
2. Bear spray is sort of like a flame thrower, it makes animals go away and it doesn't require high precision to apply
3. If you still feel you must carry a handgun it should be light, handy and accurate. Since I think you won't see any lions to shoot, I think you can at least amuse yourself by shooting some pine cones, tin cans and empty shotgun hulls you find of a log before packing them out.
4. If you live in grizzly/polar bear country, you know what to do anyhow.
I like my little 3" barrel LCRX 38 Special. It weights one pound, unloaded. Its for feral dogs, sick/wounded animals found afield, but in reality it's shot many a pine cone. I just drop it in the pocket of my jacket.
When I feel like a cowboy, I take a Ruger Single Six .32 H&R magnum loaded with wadcutters, in a western style holster. Even loaded light, it's no joke. It's accurate as a laser beam and among my crowd far more status than the latest biggest iPhone.
Let's just come out and say it- it's fun to carry a pistol in the woods. It doesn't matter which one you have, they are all good enough.