Hunting caliber for close-long range ?

Kind of like Jack, in Eastern WA now after growing up in Idaho and 30-06, 270, 308, etc are all fine; you could look at the 270 WSM or Weatherbys but really don't need to be that overbored. You can get a decent little Ruger American for a good price and be accurate enough towards 400 yards, but it takes handloading and work to get past that with any consistency and I don't lob stuff out that far. I just figure my woods/hunting skills should get me close enough to match my shooting abilities.

As for the 338, it will work better with mid to heavier bullets for the cartridge, thinking you might have gotten it for elk, but it isn't the cartridge and weight of the bullet(s) it's the bullet terminal performance profile. I use Barnes TSX in the 300 WSM and Nosler Accu-Bond in the 30-06 my son uses, both 165 grains, and would use the loads on deer, elk, black bear, and moose if I got a tag because I trust the expansion dynamics of those loads to be reliable in every case. There are some very good "premium" loads available in factory ammo if your rifle(s) will shoot them well enough, I've been using controlled expansion and bonded core bullets for years now and never had a single problem with them setting up when you hit the target. Accuracy lets you choose your target better too, I'm sub-MOA on the loads and I take chest shots whenever possible just behind the front leg and 1/3 the way up. That takes out the heart, lungs, aortic arch structure, and diaphragm when done right, and the animal isn't going far. 

And like Jack suggests, set up to be 2" high at 100 yards and be ZERO at 200, that way 300 is a lot easier to calculate and 400 is a hold on the spine or just above. But as I said, I'd rather work to get to 200 yards given the choice. 

Don't shoot the meat = no wasted meat.  Your Lapua will do just fine when fired through the lungs, heart, or neck of deer.  Choose your ammo based on what your rifle shoots best, your particular gun may not group well with the lighter bullets, and learn to pass up shots if conditions aren't suitable.

All of the other calibers you've listed will serve your stated purpose.  The availability of guns and ammo is always different in different places, and I think this is even more true right now during the pandemic.  Locally here, I see more 6.5 and .308 ammo on the shelves (when there is any ammo at all), but that might not be the case where you are.  It'd suck to get a bunch of recommendations for a particular caliber and have all of us telling you how awesome it is, only to find out you can't find a gun or ammo to shoot in it.

I have used a 270 With 130 grain bullets. It works well for deer and really reaches out there. I like it as it does not ruin much meat. I have shot many big deer and a elk with it as well. A great all around caliber.

I have hunted Mule Deer, Bear and Elk with an .06 for about 70 years in Eastern Washington where we sight in at 200 yards and hold the same POA out to about 250- 300 and it has worked very well for me over that time. All of those you mentioned are good for deer. Hang a list of them on the wall, throw a dart, go to the local gun shop and get your rifle.