Luigi Barbasetti wrote two books on the art of fencing; it is doubtful he wrote a widely accepted code on the rules of duels, especially dueling with hand guns.
A title card reads "a prince has a duty to his country higher then [sic] his duty to himself" - a grammatical error unusual for such a prestigious studio as MGM.
Given the common understanding that a prince has a duty to the state that overrides his duty to himself, it is inconceivable that the first and second in line to the throne would consider, or be allowed to fight, a duel between themselves.