You can't fight anyone, that's the challenge. Use the vent in the bathroom to get into the next area, then use instinct to spot where the fights are going on and avoid them by walking past, make sure you're far enough away that you won't get sucked into one though.
Started a fight myself in the second area, snuck through it without starting a fight and still nothing, I can't start a fight in the first area because I find it impossible to get to the bathroom. --- GT:Emotional Hobo
Switch off the jukebox in via the switch in the bathroom, enter the vent and go into the next area. Avoid any fights there, it's easy enough to do on Hard. Then simply walk up to the bartender and you should be done.