I think that is a bit doom and gloom. If dueling persists or is reconstituted I would expect it to be used largely to settle personal or civil matters. Also just because a matter is settled on a personal basis doesn't mean the government suddenly decides to stop prosecuting a criminal unless of course he is dead. You don't have to accept a challenge by the way. You can tell them to piss off and the only loss will be in your position in the eyes of polite society.But what does this mean for the justice system? We're talking about an appeal to something other than government-backed justice. That's such a major departure. It would definitely undermine legal authorities. Probably most noticeably in cases of sexual assault. A lot of angry brothers and fathers are going to mean these situations are rarely prosecuted in the criminal courts.
Or the courts adapt and consider honor as integral to the law. More eye-for-an-eye punishments, perhaps? Forced marriages to couples who conceive out of wedlock? And probable chattel-hood for women.
Ok after spending a hour trying to come up with a passable explanation for honor and the legal system let me sum it up thus. If they think honor should be part of the legal system it will be. If not it wont. This will happen largely regardless of the existence or non-existence of dueling but on wither honor is a personal or communal commodity and responsibility.