I use winchester wmr in mine and get 5-8 fps extreme vel spreads. The 200 AB tops out at 3400 fps and the 250 AB tops out at 3125 fps. These are the two I have with 26" barrels. Virually identical to my 338-300 ultramags with 28" barrels and about 75 fps faster than my 340 Weatherby with a 28" barrel.
Wondering if your rifles all share the same receiver-magazine oal, LTLR?
How much better performance can/could you obtain if you were seating your bullets in the smaller capacity cases to the same oal your .338LM allows? Have you looked at the .338 Norma which is supposed to be in the same ballpark as all others you mention, including the LM and 338-378; but being several tenths shorter enable optimum seating flexibilty?
With the super-mag cases, you lose much seating variability, unless your receiver has a 4" magazine oal. The Sako TRG42 or TRG-S with M995 action has 3.75". I'm not familiar with Lawton or Stiller super-mag sized actions but how many are using them with anything but the largest cartridges anyway? Ultra Mag and 340 Weatherby are all standard magnum functional.
My point is without having to stuff half the bullet below the case neck, likely you would get same neighborhood performance as the lapua and do so while burning less powder or using those with less than slowest burn rates.
Just happen to have the Accurate Arms #2 by the desk and it shows in a .338RUM that a 250gr Grand Slam driven by 79gr of xmr4350 yields almost 2800fps, while same bullet w/104gr of their 8700 yields 25fps LESS! Not debating that there aren't more energy producing powders, but if you had an extra .25" (or more) to move the bullet forward, freeing up additional capacity, should be able to boost those velocities.
The same load of xmr4350 in the .340wby yields the same velocity as the .338rum according to AA, but at only 61000psi. With another .15" of seating depth and throat optimized for it, in the .340wby, and going up to 65kpsi you could be in the same ballpark as the Lapua but with 20gr less powder...
Just some thoughts.
Now I will shamelessly plug the .338/300 win mag which I've embraced because it allows more flexibility than any large magnum. From 65gr of xmr4350 to 79gr with a 250gr bullet; plus oal case length of 2.62" which allows seating your bullet almost flush with the neck. About the same performance as the .340wby, but with wider variety of brass and lower expense.
I understand about Lapua brass, and they no longer make win mag brass, but in a 14lb field rifle such as a .338/300 win TRG42 you won't "need" a brake. With more mid-sized ctg case you don't need the 28-30" barrel to burn all that slowest burning powder.
Of course the .338LM really needs a 4" minimum oal magazine to get all its horsepower and torque working for you. Yet, those actions are really heavy, or I would expect them to be, and the military shoot factory loads to specs anyway so the magazine oal is not valued.
Custom rifles, chambers, wildcats and handloading variables are what this a most fascinating pursuit. When it comes to the power element of highpower rifles, nothing can do as much as a .338 bullet.