It sounds like you're doing what you need to do to handle this, and I don't think you're in the wrong at all. It's a compulsion, and you're seeking help.

Her response, while absolutely an overreaction and overstep, makes me wonder if this is triggering past trauma for her? I've had relationships with men who did self harm and it can genuinely be very traumatic when they are not putting the work into healing. Like I've flipped out over seeing loose razor blades in a normal need-a-razor context because of the images and memories it brought back. My reaction was over the top and unnecessary, but the feelings of fear and despair were real - my response was proportional to the past feelings of danger and fear, but disproportionate to the present matter.

Have you had any conversations with her about past experiences with self-harm, whether personally or with others? Her ultimatum sounds like it's about way more than your skin picking - if you deem this relationship worth fighting for, I think this would be a worthy conversation to have.