Both the 9,3x62 and 375 H&H will serve you well. The area / terrain in which you will be hunting may be a factor to considder.
If you are going to creep around after bushpig or bush buck, the 20" CZ550 Full stock may serve you much better than a 25" BRNO or 26" Win Alaskan in 375 H&H.
Load the 250gn Barnes for both calibres (T TSX in 375) and you will be able to take almost all animals.
Do not overlook the great 375 Ruger. I have owned and hunted with a 375 BRNO 602 in South Africa, and have owned a great Tikka T3 in 9,3x62 here, but have moved on to two 375 Ruger rifles (one African, one Alaskan), and believe this compact but powerful combination, offers you the besy of both worlds.
If recoil may be a factor to considder, go for the Ruger Hawwkeye African or CZ 550 in 9,3x62.