I wish we could see him in a duel. Remember that mission at the war table, where someone wanted a duel with him? I would have liked to see him fight, or making a report about how it was, but he sent someone to fight in his stead (I was surprised, I thought he would go...) but yeah, losing our military advisor in a little duel at the beginning of the game would not be a good idea
Cullen couldn't go himself on that quest, because the duel was a trap of the Orlesian Game. The Vicomte accused Cullen of being personally responsible for the death of his brother at Therinfall, and whatever move Cullen made in response to that would be incriminating himself - if he accepted the duel, he would be acknowledging that he was personally responsible for Lord No One Care's death; if he refused, he would also be acknowledging that, but at the same time running away from responsibility, marking himself as a dishonorable man (and thus one that, according to the Orlesian nobility rules, is an open target for the Vicomte to take out in any way he wants, no code of honour necessary).
So you have different responses to the challenge, with Josie getting the guy to calm down and Leli pulling some strings in the background, but Cullen's is actually the most appropriate response according to the Game: by sending a duelist in his place, he acknowledges his professional failure as a Commander in protecting the civilians who were killed at Therinfall (instead of his personal failure as a single man); but also denies the Vicomte, and the Orlesian nobility in general, the satisfaction of seeing the Inquisition's highest ranking commanding officer reduced to petty squabbles in the streets. Whatever happens at the duel in inconsequential at this point, Cullen already won the Game by accepting the challenge, but not deigning it worth of attending in person.
(This is the kind of brilliant stuff hidden in the frays of the writing that makes me think that, for all that Cullen despises nobles, he can handle them and their games just fine. Part of being Captain and later Commander in Kirkwall was dealing with the annoying nobility there, and he had years of practice handling pompous a-holes whose donations the Order and the Chantry depended on. He obviously suffers from deep social anxiety if placed right inside the viper's nest and surrounded by them like vultures like at Celene's ball, but if forced to play the Game to preserve himself or the Inquisition, Cullen can and will play it as well as Josie and Leliana; if not even better, as in the case of this quest.)