I disagree with your definitions. I would say justice is sanctioned legally, while revenge isn't. Aside from that, they're nearly indistinguishable. And given what is and isn't legally sanctioned by countries ("torture" for example), I think the difference is even less.
Sure, justice gets twisted on a regular basis. But that doesn't mean it doesn't exist, as a concept, as something different from revenge.
Also, they already released all the info themselves, so Barr giving it to the police would be redundant.
Regardless, poetic justice is more about fate making the good guys get their reward while still being good and the bad guys get punished as a result of their actions. This isn't a case of fate. It's a case of someone (deliberately) posting someone's social security numbers online for everyone to see. Was that the just thing to do? I suppose that's debatable. But I don't think it meets the literary definition of poetic justice.