I would have to say that the protagonist of the story (the barber) shows a bit of each in this story. But overall, I think it's more of cowardice.
You could say that he is showing courage because he refuses to kill Torres even though he would like to. You could say this shows him sticking with his morals rather than becoming a murderer like Torres.
But on the other hand, you can say that it was just cowardice that made him refrain from killing Torres. He thinks about how much trouble it would be and what might happen to him if he did.
The last line of the story implies that maybe he is being a coward. Torres says that killing is hard and the barber wasn't able to do that difficult thing.
The Barber's response in the short story "Just Lather that's All" is more complicated to make him just a coward. Cowardice would mean that he was too afraid to take action. Yet, he desires to take the action but thinks better of it. In some ways not responding to a situation is more courageous than making a foolish choice. If the barber were to take action and kill the man, he would not stop the man's men from coming after him and the others. In his mind their cover would be blown. He and his comrades might be more valuable, if he does nothing. However, he is not aware that he man already knows of his identity.
The barber looks at the man and sees the following:
"But the spectacle of the mutilated bodies prevented me from concentrating on the face of the man who planned the whole thing, the face I now held in my hands."
This sets in motion his strong desire to end he man's life. He also thinks to himself:
"A lot of our men had probably seen him come in. And having the enemy on your turf makes you act a certain way."
In the above passage he may be aware that he is expected to do something abut he situation. He then states later the following:
"It was going to be very hard, then, to explain that I had him in my hands and I let him go, peacefully, alive, and clean-shaven."
This statement explains that he knows his choice not to kill the man is not going to be a popular one with his peers. In the end he realizes that it would have been easy to take the Torres' life, but he does not want to kill. It is not his nature. He is not like Torres. He is a barber not a murder.
"But I don't want to be a murderer; no, sir"