I have used a .375 exclusively on buffalo with great success. Any second shots were purely insurance shots. But let's say a buffalo is inbound with the intent to do serious harm. I would argue a .375 is just as likely to terminate the excitement as a .416 class rifle. Neither bullet in the chest is probably going to stop or divert a charge at close range. A 300 or 400 gr bullet going up the snout will. I simply believe most people are likely to put that bullet into such a small target with a .375 than a forty something.
I will agree that a .458 Lott and the 500's do have the potential to turn a charge, but even with those, I would be trying to get 500 grs of whatever into the control center under the boss.
So, if I have a choice between a .375, a forty, or a .500, I'll gladly opt for the .375 and its flexibility and ease of carry.
Back to the OP's original question, I am certain that a 250 gr bullet of the proper construction from a .338 would be absolutely lethal on cape buffalo. After all, the 318 WR was a favorite for buffalo (and rhino and elephant and everything else) for the better part of two generations before and between the wars. That said, a .375 is a very sensible minimum.
Should you wish to try something smaller, book your hunt in Mozambique where there is no minimum at all.