Oh dang it. Can't sleep. Brainstorm raging. Wall of text incoming! Spoilers abound!
We know Josephine is a master of the Game. She is a skilled diplomat. She hates resorting to violence. She's also a terrific Wicked Grace player (significant, bear with me).
How did Josephine become a diplomat? Well, it goes back to an incident in her youth. She had a very rose-tinted view of the Bard life and thought she would enjoy it. Cue the incident where she accidentally killed another bard in a panic. There are many factors that would justify her actions, but she refuses to accept these justifications and insists it was a waste. That is interesting. Why is she so insistent on this? Here is my theory:
In this case, Josephine panicked and reacted without thinking. The consequences (the man dying) marked her, but there is something else going on here. She hates that emotion overwhelmed her and that she lost control of herself and the situation. That is what bothered her the most. Combine this with her family's declining fortunes and you have a very good recipe for someone who wants to have a firm grasp of herself and the situation.
Her skill as a diplomat comes from this: she has trained herself to shut down her emotions when she works because she knows that disaster may result if she becomes overwhelmed again. Yet she is all too aware of the power of emotions and uses other people's to manipulate them (note how she handles Marquis Durellion at the beginning of the game.)
Does that mean she is cold and unemotional? Not at all. It means she has learned to compartmentalize her emotions and she has an 'on-off' switch for them. That is why she is oblivious to the Inquisitor's advances: the switch is in the 'off' position then. This person is either a colleague or her boss. Emotion has no business being there, so it's not there. This ability is also what makes her such a formidable card player. Josephine is a think first, feel second type.
Contrast to Leliana who is a feel first, think second type (her decision with the traitor agent at the start of the game. It's a knee jerk reaction.)
This on-off switch also explains her reaction to the duel. At first she tells the Inquisitor that it's not their place to interfere. The switch is mostly on 'off' at that point (she is worried about the Inquisitor) and she is completely right: The Inquisitor has made no commitment to her, so they really have no business getting involved into her private affairs. When the 'I love you' comes out though, the switch flips to 'on' all the way and all is right in the world.
Josephine's interaction with her sister Yvette at the Winter Palace is interesting in this regard. Yvette is her little sister and Josephine acts rather irritable towards her. This is quite a contrast to her usual composed demeanor. That's because she has been putting up with Yvette's nonsense for a long time, so the switch moves faster from off to on with her.
Also, Josephine has learned over her career that sometimes people just cannot be made to agree and they can only be controlled to an extent. The only person she has absolute control over is herself, and that one time she lost even that.
So Josephine's revulsion for violence doesn't come from naivete or idealism: Violence is an ugly reminder of how she lost control of herself. That's why she hates it.