The badger fight took me a long time to finish. What i'd advise is buying a drink from the bar and then going through the events with him, that is what helped me beat him eventually. I'd tried over and over without any progress but after i drunk the drink before the fight i won first time, so though there may be a way without i think this is significantly faster
If you go downtown you should see the bar and you can access the range of drinks by going to the bar and pressing buy. Personally for myself, i bought the drink Black Horn from Grisus which is 50 money and that did it for me, but i haven't tested the other options and they could very well work too. Good luck with the fight!