Tons of options here. I would start my answer with a question: "what do you have available to you?"
For the animals on your list, most any good medium sized deer cartridge will work, the bullet will likely be a more important selection than the cartridge.
The 7.08 Rem or .308 Win will work with the right bullet, but a 30.06 will work better, especially with a good 165-180 gr (or heavier) well constructed bullet such as the Nosler Partition, Swift A-Frame, Norma Oryx, just to name a few.
If I were at all concerned I would take a magnum in the 7mm to 30 caliber range, but ONLY IF you shoot it well and have a good bullet grouping tight. Shoot everything through the shoulders as the vitals are more forward.
I worked and hunted there for 4 years, taking over 100 different plains game and dangerous game animals with .223, 7mm Mauser, .270, 30.06, 300 Win Mag, 375 H&H, 416 Rigby, as well as compound and cross bows. The critters there are built a little bit tougher, but they are far from bullet proof, and the key thing to remember is that the vitals are more forward.
The old warrior of a blue wildebeest that I hit too far back with the 30.06 took a bit of tracking to find, but the big kudu bull that took an arrow through both shoulders traveled a mere 34 steps before expiring.