Cantankerous Cops Cause Costumed Crusaders Conundrums

Sergeant Harvey Bullock was a... complicated... officer who had been a part of the Gotham City Police Department of Earth-One for an unknown period of time. While he had a spurious reputation that hinted at some unethical behavior, while led to his suspension by Police Commissioner James Gordon, Bullock was reinstated to the police force by the corrupt Mayor Hamilton Hill. As a paw of the mayor, who saw Gordon as a political obstacle, Bullock employed a prank on Gordon which backfired, leading the Commissioner to being in a coma for a short period. Regretting his prior decisions, Harvey sought to restore his reputation and became a notable office within GCPD. His relationship with the vigilante Batman was... also complicated.

On the parallel planet of Earth-Two, Detective McGonigle had a similar salty disposition towards the Batman of his world, going so far as to seeking to uncover the vigilante's secret identity. Both Batman and McGonigle save each other from threats to their lives, when both men pursue an ugly horde of gang members secretly led by Larry Larrimore, who himself disfigured the gang members as he himself had been in an attempt to seek both revenge for himself and a cure for his condition. Despite repeated tries, McGonigle is unable to unmask the Batman, and thereafter left Gotham City Police Department.

Bullock would have tense exchanges with Batman and other members of the Batman family, although he restrained himself and worked alongside these masked manhunters. McGonigle moved the Empire City, joining that city's police force where he became the supervisor of police officer Dan Richards, who was secretly the original Manhunter*. 

At this point, McGonigle gave up trying to uncover the secret identities of costumed crusaders whom he met, with Richards finding his coworker an amusing distraction from the grime of crime. It’s unknown if McGonigle came up in conversations between Batman and Manhunter at All-Star Squadron meetings, of which they were both members.

* This is confirmed in the Batman Encyclopedia entry on Detective McGonigle.