I gotta be one of the minority that are going to disagree.
I think Johnny Lawrence wins against Barnes in a death match. Johnny trained with Kreese from childhood. Kreese, being a killer, for decades had been teaching Lawrence murderous techniques that were not allowed in tournament settings. Barnes grew up training for tournament fights. And he did very little actual training with Kreese and Silver. They just brought in the best tournament merc they could find. Aggressive and mean to be sure, but he was not trained to kill.
Barnes was a better competition fighter to be sure. But in a death match, free from tournament restrictions, the true scope of Johnny's skill would stand out. And with Barnes being another evil bastard, there would be no moral conflict in Johnny, and he could go full dark side.
About 2 minutes in Johnny does a crafty leg sweep then curb stomps Barnes American History X style.