You and me both Pomoxis....
The concept of the immediate follow-up shot seems to be the clincher to the professional hunters I've seen or read about, many of whom seem to favour doubles. The idea seems to be that when they come to bear, it's a charging critter at swiftly decreasing range so they don't need a scope or the third shot so much because they only have time for two, or at least they'd trade the idea for that quick one-two snap shot right up the nose...
STOPPING cartridge? I'd be tempted to say starting at the .375, but this is life insurance... so there really isn't any such thing as overkill. Heavy side of the .400's and well into the .500's seem to be the order of the day. I've heard of a self-defense shot on a Somali Elephant with a .270 and a reversed bullet (solid-end first)... but that's a Hail-Mary play. A .223 would technically drop a charging buff right through the central cortex.... But a .505 Gibbs or a .458 Winchester would probably be a lot more assuring if you had ground-shaking, adrenaline rush, closing distance... If the bullet smashed in just off-center, it's a lot more likely to do the job enough to let you live.
Actual caliber? Like grizzly medicine. The heaviest you can accurately and physically shoot.
Theoretically for me? As in, "Welcome to Fantasy Island, Bwana Darkwolf"?
.416 Rigby double with some damn heavy solids... As expensive as a Spanish castle and almost as heavy... But tried-and-true matched with all the above would at least be the factor... and some nostalgic sense too.
Reality? A bolt-action, scoped magazine rifle in .375 would probably be in my hands at the wet-a$$ hour. I can afford it, handle the cartridge and it would be the choice for the neophyte Nimrud I would be.... which is why heavy .600 Nitro Express-double toting PH's are legally mandatory for us.
Just as a weird nostalgia question, how would Teddy Rooseveldt's .405 Winchester rate as a stopper? Or as a client's hunting cartridge for some of the african medium-heavys by the modern perpective? Realistic hunting rifle or lever-action spray-and-pray?