Your .340 is likely fine...
@Hmaxwell, the groups you produce with the .325 definitely talk to your shooting form with decently powerful calibers. This is a good base to build from.
I have had my share of zeroing issues, and I have seen a lot more with bolt-action service-issued weapons, and generally there is a fairly easy path toward resolution when one follows step by step the logic of the situation. Here, the part that originally baffled me was:
If you had not written this, I would have likely concluded that 1) the bases are loose on the action; or 2) the rings are loose on the bases; or 3) the scope tube slides inside the rings.
This last actually happened to me, precisely with the Mark V .340 Wby, and it only resolved itself when the scope slid all the way forward and the rear bell shoulder locked itself against the rear ring. Because the scope is a Zeiss, the zoom adjustment is located behind the bell shoulder, so it is not an issue. It would not have worked with many American scopes where the zoom adjustment is ahead of the rear bell shoulder and would contact the rear ring. I just could not understand what was happening for the first 20 shots or so, that were landing all over the place at random, 3" to 4" away from each others, until all of a sudden the rifle started to shoot 1" groups. Only when I cleaned the gun back at home did I notice that the scope had moved 1/4" forward and the rear bell was now contacting the rear ring, and I instantly understood the logic of what took place.
Because you say that the gun zeroes back after 45 minutes, this cannot be a typical mechanical issue. This is why I was asking in my previous post something to the effect of: can you please confirm that the gun comes back into zero WITHOUT any scope adjustment? If such is the case, by process of elimination: it cannot be mechanical i.e. sights attachment related (because it resets itself); and it is not thermal related as understood in a normal sense (1 or even 5 shots are not enough to heat a functional barrel to the point of opening a group to 12" to 14" as you stated in your first post). Dang!
But then, I look at your target, and if I interpret it correctly, your 3 shot group is NOT 12" to 14". It appears to be roughly 2 1/4" x 3." So, this it essentially 3 MOA shooting a big gun at 100 yd. Now, THAT introduces DATA, and a whole new perspective to the discussion.
If we are not discussing 12" to 14" as you initially reported, but 2" to 3", I will tell you candidly that I have seen (and I am sure @Von S. @Luvthunt and other folks on this thread have seen too) many, many shooters who could not hold 3 MOA with a .340 Wby (or a .416, or a .458, etc.) INITIALLY, even though they had experience with .300's or even .338's or .375's. It took me a fair amount of shooting with the .340 to get it to stay within 1 to 1.5 MOA, on a good day, and I do not overly worry if certain days it is 2 MOA. If you are comparatively new to it (say, less than 40 rounds shot), you are doing just fine. Keep shooting, I bet the groups will shrink. FYI, I have seen the same thing happen with folks, qualified shooters, transitioning from .300 Win Mag to .338 Lapua.
Depending on gun weight etc. your .340 likely recoils about 30% more with 250 gr bullets than your .325 does (~40 ft/lbs vs. ~30 ft/lbs), so this very likely has an impact on your groups. It also recoils about 10% faster than a .375 H&H. I am assuming that you have been shooting 250 gr bullets. Correct? If this is the case, my recommendation would be to load one box of 185 gr TTSX (or get one box from
https://hendershots.net/product/340-weatherby-magnum-extreme-custom-ammo/) and shoot them carefully. They will recoil about 20% less than the 250 gr. I would not be overly surprised if your groups start shrinking.
Just my friendly, and hopefully useful (?), $0.02
PS: the fact that your gunsmith has the same results as you do is not necessarily telling, as he may not have a lot of experience shooting the .340.