I have a 338 Allen Mag I know its possible to achieve 3300 to 3400 fps accurately with a 300grn smk. I have always loved the 240smk there has ben several world records shot with them. I was just trying to figure out how to shoot the 240 at high speeds. I also have a 300 yogi which is a 30 - 338 Laupa improved. I was only able to achieve 3100fps the extra velocity would make a huge difference. The only combination that I could come up with would be take the 408 chey tac case shorten it and neck it down to 30 cal and im still not sure that velocity is achieveable.
None of the 338 Norma/Lapua cases are going to get you there.
You will have to work with something in the 408 Chey-Tac case. But why would you want to ?
It would burn a barrel out in short order.
My 300 JAZZ(30-338 Norma Improved) shoots the 240gr SMK @ 2850fps w/ a 28" barrel, and so far has shot 1.5" groups out to 650yards. I dont see the need to push the 240gr to 3400 fps. With a 34" barrel out of a 300 JAZZ you would get to 2950-3000fps.
If you already have a 338 AM, you already have a hammer of a rifle.
Maybe you don't need to design a new cartridge. Use 308/510 sabots in a 34" 50 BMG. With a 240 grain SMK bullet and a 60 grain sabot (300 grain total) about 250 grains of VV-N560 should give about 3800 fps+ muzzle velocity at under 47000 psi (a moderate load for a 50 BMG. Thats calculate with 0.5" sabot seating depth. You will need a barrel with a 10" to 11" twist to stabilize it at sea level. The standard 50 BMG is 15". Pac-Nor can supply a 10" twist 50 BMG barrel.
A 30-378 Wby can't quite drive a 240 SMK to 3200 fps with a 34" barrel. 103 grains of VV-N570 shows 3190 fps at SAAMI max. Anything in a 30 cal bore which will give 3300+ fps is going to have poor barrel life, but maybe thats ok.
I haven't tried i either of the above loads, they're just Quickload simulations but they should be close.
You could save a lot of bother by getting some surplus 50 BMG M903 SLAP ammo. It's got a 30 cal ~355 grain tungsten penetrator and is rated 4000 FPS from a 45" M2HB barrel. It's stable in a 15" twist barrel because tungsten is much denser than jacketed lead. I've seen it for sale at gun shows a couple of time for $25/round.