In considering the whole "system" involved with a cold bore precision at 1500 yards, both calibers have capabilities beyond most hunters/shooters.
Since you state this will be a dedicated LR hunting rig for large game...these would be some I wish I would've pondered on a while back on my quest-
1. What range finder do you plan on using that has sufficient beam width divergence to ensure you are ranging the intended target vs. a bush, rock, etc in vicinity of the target at those ranges? At those ranges, given a ISA atmosphere, a range determination error of 5 yards is 6 inches high or low...which could mean the difference between a happy ending to a shot, or the beginning of a long chase.
2. B23 hit on an important point...there are many variants to the 338 Lapua Improved. In fact dare I say that throat design/barrel may account for 100 fps variance between the different designs of the 338 Lapua Magnum Improved. All things considered equal (throat design, barrel length, quick vs. slow barrel, pressure vs. brass life), a maxed out variant of the 338 Lapua Improved will have more case capacity, and will withstand additional pressure before blowing primer pockets than the 338 Edge, allowing it to outpace the Edge by 35-60 fps.
3. Ability to make consistent precision ammunition will come into play at these ranges, so brass quality and availability of "easy for you to make precision ammo dies" should come into play as well when deciding between the two.
4. The initial purchase price advantage of Rem 300 RUM brass vs 338 Lapua brass is just a teaser...don't fool yourself
You'll be so far down the rabbit hole money wise with a honest 1500 yard cold bore gun, that would be a insignificant factor in making your decision in my opinion. Once you consider brass life...this further evens the field between the two.
Go to the defensive edge website and review the load data for the 338 Edge.
I have personal experience with a 338 LMAI does 2920 fps with 104 gr of Retumbo out of a 28" barrel with very good precision.
Keep in mind most rifles shoot best short of their full velocity potential....that's why you'll see most Edges run at 2820-2850 fps, 'cause that's where they have shown their best precision potential. Same on the 338 LMAI.
Good luck and remember to keep it fun!