That version of the demo hadn't been updated since August, as I was waiting for the Steam demo release. I didn't know there was such a serious bug, though. Just updated it to the latest version, which includes some changes that haven't been announced yet. Thanks for checking it out and the feedback!
I downloaded the demo again and am still having the same issue. I even tried replaying from the beginning in case my save file was the problem, and I still get stuck in the exact same place.
...I also tried the "Beyond Drunk" game to see if that would be any different, and I CAN progress past that spot there because the barman menu does display properly in that game.