You state your rifle is intended for target work.
I think bag control of your rifle will be a major concern. A heavy weight bullet in a .300wsm is not easy to master and the 250-300 gr bullets in a .338 will exacerabate the problem.
Maybe some target shooters can address this better than I. A hunting rifle is another animal.
If you really want a target shooter perspective........
Nobody shoots the 338 WM as a long range target rifle. While it is an excellent hunting round, there are much better choices for target shooting.
I currently shoot 1000 yd BR. My light gun (17 lbs) is a 6 BRA and my heavy gun (30 lbs) is a 300 WSM shooting 215 grain Berger's. It shoots very well. I shot in three matches this year and wound up with three 10-shot 1000 yd screamer groups (4.33, 3.88, 4.9).
The 300 Norma and 300 Norma IMP seem to do well at 2000 yds, as do the 338 Lapua IMP. We have an ELR shooter in our 1000 yd BR club that is working with the 300 Norma IMP. He has also shot either the Edge or Lapua IMP at 2000 yds.
My "timber" elk rifle is a 33-28 Nosler on a stainless CRF M-70. It weighs 8.75 lbs with the scope. It shot 225 Barnes TTSXs very well at 3170 fps, but my current load is 250 grain Accubonds at 2825 fps. I tried the 250 Berger EOLs and got them close to 3000 fps, but the barrel didn't like them. I also tried the 265 ABLRs, and like everyone else, these bullets shot poorly. They are .0005" underside and we think that is why the 265 ABLRs shoot so badly in most barrels.
I have also had a couple 338 Edges that shot very well, I've had a couple 338 WMs, a 338 RUM, and a couple of 340 Wbys. All of these were custom built rifles. I have also developed loads for a few 338 Lapuas and 338 Lapua Improveds.
With the 338s there is a definite break point. Lighter weight rifles will shoot 250 grain or less bullets well with good accuracy and minimal torque. Bullets above 250 grains start to impart a lot of twist. While having a hunting rifle twist in the bags and occasionally throw a shot horizontally might be fine in a hunting rifle, that is a no go for a target rig.
@Alex Wheeler, who has tested this a bunch, says a 338 needs to be in the 12-13+ lb range to really handle the heavier bullets well.
I made my 33-28 because I thought it was about as much recoil as I wanted with a non-braked carry rifle, and 338 RUM brass was hard to find. If I were doing it again I'd just built a 338 RUM and add a brake. I needed a RUM magazine and follower for my 33-28 anyway. I would still keep to bullets 250 grains or less.
For a long range target rifle the 338 Edge is hard to beat for a standard magnum action. You can get long magazine boxes for both Rem 700 and Win 70 actions from Wyatts Outdoors. If you are not stuck on using a factory action, and you shouldn't be nowadays, the 338 Lapua IMP is excellent.
You can get ADG brass for the Edge and Lapua brass for the Lapua. And it is definitely worth picking a cartridge so as to be able to use high quality brass.
If you want a lower power 338 the the 338-300 PRC or 338-300 IMP should be at the top of your list. ADG makes brass for the 300 PRC. Don't even consider the availability of factory ammo for a target rifle.