He's a vigilante period, he wouldn't be able to testify in court even if he was best friends with every cop in town and talked to them 24/7. I don't fully understand what your point is here. You also ignored my point of how they trust him enough to install the Batsignal to contact him and Harvey Bullock who was very anti-Batman early on grew to trust and begrudgingly respect him. If Jim Gordon and even Bullock trust Batman that shows you a lot of other officers do too.