For example, what is it about the situation that causes me to try to influence Alice and Bob using moral arguments in these cases, whereas I use other methods of influence in other cases?
Guilt works here, for example. (But XiXiDu covered that.) Social pressure also. Veiled threat and warning, too. Signaling your virtue to others as well. Moral arguments are so handy that they accomplish all of these in one blow.
ETA: I’m not suggesting that you in particular are trying to guilt trip people, pressure them, threaten them, or signal. I’m saying that those are all possible explanations as to why someone might prefer to couch their arguments in moral terms: it is more persuasive (as Dark Arts) in certain cases. Though I reject moralist language if we are trying to have a clear discussion and get at the truth, I am not against using Dark Arts to convince Bob not to strangle Alice.
You’re right, I did stop reading carefully toward the end because the Kantian stuff looked like it was just a side note. I’ve now read it carefully, but I still don’t see the significance.