My favorites:
250 SGK/CCI 250/Re-19 or VV N560/2,700 fps/<1 1/2" - My go to long range load. High BC keeps it moving downrange, but moderate velocity is not going to tear the bullet apart close in.
200 NAB/CCI 250/IMR-4350/2,960 fps/2 1/4"@200 - I got to test this at 200. Noticeably less recoil. Starts about the same as a 300 WM, but will fall behind due to BC at range.
210 TTSX/CCI 250/IMR-4350/2,860/1" - Probably my elk load next time out. My best accuracy. Good BC. Less recoil than the 250s.
I'm shooting a Savage 116 "Weather Warrior". You can work out your own powder charges, but none of these are over "book" values. Have fun with the .338. You'll find it's not near as bad as its made out to be.