The OP laid out the criteria pretty well. He specified ‘bar fight’, and said “Assume the big guy is 6’2, strong, in good shape, regularly gets into bar fights, usually wins them.”
So… a small boxer, in a bar, against a fit, 6’2" guy who is skilled in bar fighting and usually wins. That’s the criteria I was using.
Having seen more than my share of bar fights, I gave the most correct answer, IMO: It depends on how the fight starts, and how mean the big guy is. If he’s an experienced, winning bar fighter, he’s going to sucker punch the boxer and use the obstacles and close-in quarters to eliminate any advantage the boxer might have. If he’s really good, the boxer will never know what hit him before the fight is over.
Let me tell you a true story: I watched a black belt in Karate get beaten up pretty good one night. He was a student from my school. A very good martial artist, and he’d been in a few scraps before. We were out one night having drinks, and he saw an altercation outside and decided to get involved because it looked too one-sided for his taste.
He went out and confronted the guy, and they pushed each other back and forth a bit. The other guy then raised his hands and said, “Hey man, I got no beef with you. C’mon, we don’t need to fight here. Let’s all just cool down.” My friend (being a decent guy at heart) agreed, let his guard down, and as he was turning away the guy sucker-punched him and immediately followed up by charging him and taking him down. My friend had the wind knocked out of him by the hit and collision with the pavement, and probably cracked his head on the sidewalk too. The other guy used the opportunity to box him about the face and head for three or four shots before his buddies hauled him off. Luckily, my friend was just dazed and bloodied up a bit - no permanent damage.
That’s what real streetfighters do. They don’t fight fair. They don’t square off against you and put up their dukes. Real street fights are not like the brawls you see on TV, and they’re not like the schoolyard fightfights you might have seen as a kid. They’re fast, they’re dirty, and they’re vicious. If you’re not the vicious type, you’re already at a huge disadvantage.
My friend learned a valuable lesson about the limits of training when you’re not really the street-fighting type. Or at least to never turn your back on someone who was ready to fight you moments before.