You can fight him as many times as you want. Its best with 2 people or more have one person stay at the entrance to the bar and travel to the beatdown when you are done. After pete is dead u can use an elevator to get out of there solo without dying but that puts you at a distance away from the bar so its annoying to have to run in a big circle every time.
I always thought you needed to travel out of the bar to get him to spawn again I tried to do it a couple times just leaving the arena and coming back but it just took my eridium and didn't spawn him. Could have glitched tho sometimes he doesn't jump down for some weird reason. gearbox is pretty legit at programming a buggy game --- --- xbl gamertag: xAbsolut Zero