Mirumoto samurai do Iajutsu duels the same way as other samurai
Nitten duels are duels of kenjutsu, not Iajutsu, because Nitten style itself is a style of kenjutsu. Iaijutsu duels are (almost) always performed in the way you described, that is the way marked by tradition, and the only type of duel allowed for serious (read: officially sanctioned) duels. Exceptions are quite rare (certain duel in which certain crab used a tetsubo in a Iajutsu duel was a one-time occurrence that will probably never be repeated). So only when the duel is about kenjutsu, or it devolves into a skirmish, then they defer to their unique style.
Note that is no wonder that the traditional/official way of doing Iaijutsu duels favors so much the trademark style of the crane clan. After all, the creator of the style was the first emerald champion and his descendants had a overwhelming influence in the imperial court and in what was considered customary or not since the very beginning of the empire.
Also note that the fabled rivality between the two schools is not as much about dueling prowess and style as it is due the philosophical and conceptual differences between the two styles and the rivality between the founders of those styles, Kakita and Mirumoto.