The .338 Edge is an excellent choice, especially with a 26-28" barrel. However, a suppressor will make it even longer, so it would help if you considered that. Also, to take full advantage of the long, heavy, high BC bullets, you need nearly 4" COAL and some modifications. Yes, a .338 RUM and a shorter barrel for suppressed might be a better alternative. A competent and reputable gunsmith should be able to open up the bolt face or replace it with a Lapua magnum bolt face. Whichever route you take, ensure you have the right barrel twist for the bullets you will use.
I have a .338 Thor (NMI—
https://www.longrangehunting.com/threads/my-338-thor.327058/#post-2776837) with a 24" 1:8" CarbonSix barrel and custom contour (.860" at the muzzle). I have an EC tuner brake and a Rex Silentium suppressor for it. Good luck with your build.