The M-14 design is inherently sensitive to pressure changges.
The powder/bullet weight balance in an M-14 is optimised for the correct port pressure required to operate the op rod with exactly the correct amount of enthusiasm to reliably recipricate the reciprocating mass.
And,
not too much powder, too slow a powder, or too different a bullet weight or the balance may cause the recipricating mass to recipricate too quickly, and beat the snot out of your receiver.
While I've never personally bothered with one, there are adjustable gas plugs available for the M-14, to bleed off excess gas. These are mostly used by 1000 yard shooters, who require a heavier bullet than the recommended 168 Gr maximum.
If you do get a barrel chambered in .300 WSM, you might also be looking at feeding issues as the mags and feed lips might need to be tweaked.
Personally, if I wanted to change the ballistic profile of an M-14, I'd be more inclined to look at .243 Win or .358 Win.
Or else I'd be looking at an AR 10, which has a wider tolerance to pressure, and probably would be a better starting point.