Are you asking if using the sword would get you a second stacking of Mountain Rage? All it would do is get you the normal benefits of a regular Barbarian's rage. Most rage effects stack with eachother, unless there are typed bonuses (like morale bonuses to will and such). So you'd Rage (mountain rage) gaining +6 str, +4 con, and Large size, +2 to will, and -2 ac., and when you use the sword in battle, you'd get another +4 to str and con and another -2 to AC. Just like going into Frenzied Berserker (including the inability to stop attacking things, even your allies), this works like stacking Frenzy (or other similar effects) on Rage.
Unless I'm mistaken, in this case you'd have a cumulative +10 Str, +8 con, +2 Will, -4 AC, with large size. You wouldn't get any mountain rage enhanced bonuses from the sword, such as the extra +2 str or another size increase.
You also can't use the sword to trigger Mountain Rage. If you used just the sword without also using your class-granted Rage, you get just the normal effects of regular Rage (+4 Str and con, +2 wis, -2 AC), plus being unable to stop raging and attacking creatures until you or everything else is dead.
While Racial Emulation works fine on paper, make sure your DM is ok with it, because otherwise they've got full rights to call shenanigans on this.