Unless you have other plans later for the 338 I would stick with a 30 cal. Anything from the .308 Win, 30-06 , 300 wm, 300 PRC, or my favorite 300 Wby Mag. The logical is the 300 WM as ammo is easily available.
You can do it all with a 7mm also. The 338 isn't large enough for dangerous game and not needed even for the largest plains game like eland or tough animals like zebra and wildebeest.
Also the 200 yard zero is antiquated. Put a scope with a dial turret and sight dead on at 100. If you want a 200 yard zero then dial that and put a small tag on your scope as a reminder. 40 to 350 is common shot distance.
Use quality bullets that are either Barnes type or bonded core. African animals are all tough. Shoot them through the shoulders with a good bullet.