I'm not sure your stated project fits in the arch typical "mountain rifle" (fly weight) category but I sure like the sound of your components, including cartridge selection.
Honoring your stated criteria,and preference for RUM's then a300 RUM, hands down.
If you feel it necessary, then you can always throttle backthe loadsa little.
A 375 RUM while a great cartridge, generally falls outside most folks mountain rifle "model" based on associated weight.
A 338 RUM is too close to your existing 338WinMag (though it is superior).
A Super .30, like the 300 RUM, should be a nice addition and fill an empty spot inyour rack of rifles and should be superlative for goats.
This all presumes that you are not infected with the recoil sensitivity disease and know the fundamentals of proper recoil management versus relying on the old "school of hard knocks" (literally) and theall too typical result of "wondering what happened."
Footnote: the aforementioned 7mmSTW is a worthy consideration and has the wonderful bonus of not being a magnum.
Again, though, it is not what you asked for.