Why are you giving up on the 375 cal. Not saying its better but the 350 gr SMK will do anything the 300 gr SMK will do in 338 and do so at same velocity with larger frontal area and higher kenetic energy payload.
Also an increase in recoil but there is always something that has to give.
The 375 will also have a much longer barrel life. Not as many bullet options but still a good one. Either will serve you well so do not run from the 375 just because there is only one bullet. If you do go that way, make sure you get a large enough supply that you need not worry about running out.
In my 40" Black Sunshine in 338 AM, I drove the 300 gr SMK to legit 3500 fps velocity. In fact they averaged 3510 fps. This was with the old TTI brass and it offered pretty good case life, 4-5 firings at this level. The new Jamison brass will not do this but it will do 3400 fps easy enough.
In my experience, accuracy generally stays at or very close to the 1/2 moa range but once the barrel goes, it goes quickly. In fact dramatically noticable in the three barrels I have burnt out.
My rule of thumb when setting a barrel back is that you cut off the entire chamber including throat and then rechamber. In the case of the 338 AM, you will clean up roughly 4" of barrel length which mostly will clean up the severe heat cracking but you will still have some.
I have also noticed that if a virgin barrel will get you say 800 rounds of accuracy life, when you set the barrel back and rechamber, you will get around 50% of that original barrel life so expect another 400 rounds before things really start to fall off. Is it worth it, your call. The damage is down the entire bore, or should I say wear. This is why the next round will not have as long of an accuracy life as the original chambering.
You may possibly get into the 14 lb range with a repeater but it will not be easy. You will need a carbon fiber stock, slim barrel contour, likely no longer then 30-32" and you will still be pushing things pretty hard.
Its MUCH easier to get there with a single shot. My BAT based Xtreme Heavy Sporters can start at around 16 lbs ready to hunt with including a NF NXS 5.5-22x 50mm scope and harris 9-13 bipod. Its hard to get much lighter then this and retain long range consistancy. These rifles really put a strain on a rifle system. They need the receiver, stock and barrel integrity to handle it and remain consistant.
The Chey Tac rounds are the smallest capacity of the group and Chey Tac loads them pretty mild.
The Sniper Tac, Extreme and Big Baer are all very similiar in performance and roughly 100 fps step over the standard Chey Tac. They are also very easy to work with as there is no extreme case forming, just load and shoot.
My 338 AM is the largest capacity of the group. Shoulder location is moved forward to stretch the body section and increase case capacity. This can do a couple things for you, give you around 75-100 fps more velocity on the top end with same bullet and barrel length, or match the slightly smaller 408 based chamberings with a bit less pressure which will offer longer barrel life to some degree.
From the low end to the top end, there is only 200 fps difference between them all and from what I have seen, all are extremely accurate at nearly any range your bullet remains super sonic so you could really not go wrong with any of them.