You said not to punish them. How do you intend to teach them empathy if all they learn is that they can get away with doing what they want?
Because punishment shouldn't be the focus. It shouldn't be "you did bad thing therefore bad thing happens to you". I'm not saying it has to be a totally positive experience for the bully, bully might even have to get their teeth rearranged for them, but what I take issue with is the institutional, top-down, carceral mentality of punishment. The focus should be on making the victim whole and making sure the bully doesn't bully again, not that the bully gets some requisite amount of pain and suffering.
The adults who focused their ire on me anytime in stood up for myself, instead of having my back or telling the bullies to knock it off. No, they just told me I'm supposed to be better than them, so I'd get in just as much if not more trouble than they did, because "held to a higher standard" or some shit.
Yeah I'm absolutely not saying that. If you beat the shit out of someone bullying you, you are generically in the right as far as I'm concerned.
So no, don't let bullies off the hook because you think "punishment" is a naughty word.
Again, that's absolutely not what I am advocating. Quite the opposite, actually. I am advocating that we treat bullies like people capable of change. Hold them accountable for their actions. And if that process is painful the bully, so be it, but inflicting pain and suffering on the bully should not be the primary goal. When we do this, the bully simply learns to pick better targets.