Some advice I got was to stick with calibers that are popular in Africa just in the odd case your ammo got lost in transit.
30-06, 300 WM, 300 H & H, 375 H & H are all very popular and most outfits would have ammo at the lodge or able to get it. I suspect they might have 338 as well but just never paid attention to that one. As mentioned, there are some quirky rules so best to avoid all possibility of problems. Personally I would go with a 300 win mag or 375 H & H unless you have a specific desire for a 338. Unlike a deer camp in the States, no one will ridicule you for shooting an Impala with a 375. They do it all the time. And with lighter bullets, you can get ballistics similar to your 30-06. I used Hornady 270 grain SP RP Superformance shells with great success. My wife shot a 30-06 with 150 grain GMX and her animals got just as dead. We both had 80% one shot kills and that 20% was shot placement, not caliber.
If you have all those you listed in a bolt gun, pick the one you shoot the best and like the most. But if you are buying one, ask yourself if you will ever shoot a buffalo or dangerous game that might require a 375 as a minimum. I bought a new Winchester model 70 in .375 H & H and love it, it is accurate and I'm covered for the future.
I had only auto and pump rifles so had to buy new guns for Africa.... The rest of the story is that you may get people inspecting your guns at the airport who don't know what they are looking at. They might think a pump is an auto. So keeping it simple and common is best. We had two people in two different airports ask to see and confirm we had the bolts removed. And they confirmed the serial numbers on the guns AND the bolts. They are used to looking at bolt guns.... You can take single shots and doubles as well, and lever actions. A simple little thing is to have your gun laying in the case with serial number up, on the gun and bolt.... Makes it a lot easier to deal with when you are in a busy and cramped space showing an inspector on a little shelf and trying to keep from dropping everything over the edge. Common sense but I never thought of it and no one warned me.