In conflict, violence is always incidental to intent (unless your intent is to hurt someone badly). It's a method of getting what you want. Maybe you want someone to admit to his wrongdoings, or maybe you want him to keep his mouth shut, or maybe you just want to take his sandwich because you're hungry.
You could do all that without resorting to violence. It's just that violence is a shortcut. It's faster and requires less work. In RPGs, violence is often applied as a blunt instrument, but it is best when used as a scalpel.
Moves model intent. Why not make violence incidental to moves themselves?
When you resort to violence to get your way, inflict harm as established. You might also suffer harm as established if your victim puts up a fight. Then make your move and add the following choices:
• You roll +hard instead of anything else.
• You take +1 on your roll.
• You inflict terrible harm (+1 harm).
• You suffer little harm (-1 harm).
This might not graft entirely cleanly onto the system. What are your thoughts?