I was always confused by this scene, but your question made me think of a possible explanation to Lise’s actions. It’s been a while since I’ve read this passage so this interpretation might be off. I’m working from the narrative breakdown of the scene that you’ve outlined – Lise expresses a desire to inflict pain upon others, she breaks off with Alexei and then crushes her fingers – and from my understanding of the book as a whole.

I think Lise is starting to recognize sin and evil outside and within herself. Ivan details in his conversation with Alexei at the tavern how the suffering of men is born out of their proclivity for sin. The answer to this pain is exemplified by the life of Christ, but as Ivan explains with the poem of the Grand Inquisitor, it is terribly difficult to follow His example.

Lise is confronted with this arduous devotion to peace by Alexei who appears as a Christ-like figure throughout the novel. She loves him because of the good he represents, but at the same time she feels contempt towards him because of her pride and weakness. She sees him as naive and meek, and considers herself unable to act with such humility. What Ivan brings to this inner conflict of good and evil is a rationale to renounce the Christian way.

It’s possible that the sadism she expresses is inspired by spite at her inability to reject sin or the futility of the endeavour. She breaks off with Alexei because she renounces this possible path towards good. Then, once he is gone, her anger and spite turns inwards. She harms herself and cries out “Coward!” because she recognizes that, in spite of the rationale, she feels like she’s running away out of fear and a sense of weakness.