The OP is asking about a "Mountain Gun". That rules out a lot of guns guys are talking about here. Leave the 380's and nines at home. Even the snubbie 357 is iffie.
Stopping a 4 legged critter is different than stopping a 2 legged critter. Rifles are different than handguns. Handguns don't reach the velocity that rifles do so cast boolits and jacketed bullets act differently in a handgun. Wide Flat Nose or Long Flat Nose cast boolits are intended to poke long, straight big holes in lots of flesh and bone and retain their shape. Jacketed hollow points (bonded or not) are designed to expand and stop inside the body. For 4 legged critters we want the boolit to penetrate hide, muscle bone, vitals, etc. and make an exit hole no matter what angle it hits the critter. Granted, the hollow point in the PROPER load in the PROPER handgun with the PROPER shooter can kill thin skinned game like say whitetails or even a blackie. But, he usually has the luxury of shot placement. In a defensive situation with something that can bite back, we will probably not have that luxury. The LAST thing we want is to shoot a bear, cougar, lion, etc. in the shoulder or chest with a 230gr Golddot 45. (I do however carry those in my social gun). If the boolit hits the shoulder say at a quartering facing shot, we want it to bust the length of the critter, take out the vital and who know, maybe even bust out the far side hip joint while it's at it. That ain't gonna happen with a hollowpoint. That's a job for a heavy hard cast boolit.
A handgun is a drill, a punch press. The typical handgun cartridge from the 357 up to the 500 Linebaugh and between will make a long, straight hole if one shoots a "heavy for caliber" cast boolit of the proper design. These work best when shot at 1100-1400 fps with 1200fps being the "standard". But, it depends on the caliber and the weight. A 525gr .512 boolit (500 Linebaugh) at 1100fps will out penetrate a 360gr .452 shot at 1400fps. It all comes down to physics. A BEEG HEAVY "SLOW" boolit will do the job. In the words of John Linebaugh… (paraphrased) "Caliber and boolit weight are the only constants. Everything else is a constantly diminishing variable" like velocity and a hollow point that sheds weight.
I'll not get into the 460 and 500 Smiths. Since they are X-Frames one could argue that they aren't really a "Mtn gun". Definitely not a gun you're gonna pack on your hip for very long. But, that's a whole different thread. They have thier place but don't seem to fit in this conversation about a backup "Mountain gun"
The breakdown is something like this for "heavy for caliber" cast boolit.
357 mag - 200gr
44 mag - 300-320 gr (depends on gun if it can handle longer boolits)
45 Colt (Ruger only loads) and 454 Casull - 350-360gr
480 Ruger - 420ish
475 Linebaugh 450ish
500 Linebaugh - 450-525
Any of the above boolits at 1100-1200fps will do the job. 454, 475L and 500L will push the heavies to about 1400. Remember, if the boolit hits the hide at .452-.510 there is no need for it to expand to make a larger diameter hole. It's already as big as an expanded 30 cal. In fact we don't want it to expand, we want it to hold it's shape and weight and penetrate. If we push them too hard there is a point where there is less penetration at the higher velocity. Again it depends on boolit design… nose profile and hardness. Don't get hung up on velocity with handguns. It translates to higher pressure, recoil, etc. and doesn't do much more than kick in the testosterone. We're not dealing with a rifle.
Here is an example of a boolit I cast that I shot from my 4" 454 Redhawk (custom conversion from a 45 Colt). This load is just under 1300 fps, 355gr, hit the steel at 25 yds. Made a nice hole. I can shoot this target all day long with 45 acp 230gr hp or 200gr SWC's and they just spin the target.
I shoot the same boolit in my 45 Colt Ruger Super Blackhawk Hunter at 1200fps for hunting. I've been trying to connect on moose or bear. Came close but still trying. This load will go through 36-48 inches of critter easy.
Bottom line is pick a gun/caliber that can shoot at minimum, a 200gr hardcast at 1100-1200 fps that you will pack and that you can shoot! Shoot it a LOT. Get what you want as far as SA, DA revolver or an auto. Whatever it is, shoot a few hundred pounds of cast boolits in it and then you might stand a chance of stopping that charging critter that wants to eat you alive…slowly… you know… the ones that are out there…waiting…..watching.... ready to pounce....
For most guys the first shot will probably be a miss but the noise and muzzle blast will probably be enough to turn the critter.
In the lower 48 I'd probably pack Ruger blackhawk 45 Colt with a 1200fps 300gr load or a SP101 357 mag, 4" barrel with 200gr hardcast. If I were gonna go with an auto probably a glock 10mm with the right load. Close to a 41 mag but not quite. But I don't care much for brass flingers.
Took it sheep hunting last week but could'nt close the gap any more that 277yds. My buddy took the ram with his 270.