The Avengers bloody rampages

I would say it's more like the GCPD decriminalized Vigilantism like some cities, counties and states do Pot. What he does is still highly illegal, but they don't enforce it. Which brings up weird precedence for evidence and Batman's crimes. If he does operate like you say under the jurisdiction of the GCPD all evidence he collects illegally wouldn't be admissible in court. If he is an agent of GCPD he would need warrants to enter places. No chain of evidence is kept etc. There would be more wiggle room if not seen as a agent of GCPD if his interactions with criminals is also seen as a crime. If one gang in their commission of a crime exposes the evidence of another gang's crime then all is good. So legally GCPD would still have to classify Batman as a criminal not part of the GCPD.