'Caliber' is something that hangs a lot of folks up and can be confusing.
There is bullet diameter and speed. Put more powder behind a bullet and it goes faster.
Decide how far you want to shoot and then decide what diameter bullet and how fast you want it to go for that range. It's too easy to get caught up in the new stuff and marketing, but all you need is an accurate rifle and good skills and you can get a lot done.
I'm with 26Reload on this.
I would call a .270 or even a .243 plenty of gun for mountain game. A friend of mine has hunted all over the world with his Winchester 100 in 308. He has all kinds of critters in his house. A tiger, grizz, dahl sheep, all kinds of deer, you name it. He is old and doesn't hunt much anymore. He'll tell you how far a 308 will kill big critters, and it's a lot further than even I thought. He never used a fancy reticle or balistic software, he just shot. He got a big whitetail last year and the shot was a feat for someone in his 80's. The buck was chasing a doe and took off running through the woods. That man drilled the buck through the heart and both shoulders and dumped him on the ground with only a second or two available to make the shot. He says the 308 is a bit much for most game, but it's what his favorite gun is chambered in.
Bigger calibers are preached like gospel, but in most cases they are overkill and not needed. People have been shooting big game for years with everything from .224 to 30-06. When I was younger the 30-06 was big medicine for anything you could hunt and anything bigger was exotic. Now the advertised standard is 300 mag or 338 or even 50 bmg and I don't understand why. If you need a bigger gun to increase the chance of a kill, you need to get better at shooting. It won't be long before 20mm is the new big game standard.
If the 270 is too heavy, get a lighter stock and head out.