The best and least expensive way is to buy a rifle in 338 WM and just have it rechambered. I just did this with a new Win Supergrade and ended up with a beautiful rifle that shoots excellent. Also no bluing or other messing around, knock the barrel off and in 1/2 hr you have a 340 Wby. I've had 340s for many years and it is an amazing cartridge, I really recommend a 26" tube though, to realize the full potential. This was why the Supergrade caught my eye, available in 338 AND a 26" tube. I make all my brass out of readily available R-P 375 H&H brass, works 100% @ 1/4 the cost of Wby brass.........used to use W-W when it was available. Just got Marshal @ Matrix bullets to make me some 240 gn boattail bullets for it to try out, I think this should be the optimum weight for the 340 and have loads made up to range test tomorrow.
Make sure the action is long enough, I'm not familiar with the new Sako models but if it's made in 375 it has to be long enough, you may have to knock out a magazine block or replace the follower not sure on that action. Is your bolt throw and magazine the same on your 300 and 375, if so, you should be fine.