I do not think he is concerned with the lore as much as he is trying to figure out the different roles as presented while in the game.
The LPI are the actual law enforcement for liberty while the BH are there (supposed to be anyway) as a reinforcement group. Where as the LPI do not need to ask permission before going after a target the BH should notify the local law enforcement prior to actually pulling the trigger. Sometimes this is not possible but as a general operating procedure it should be done. As far as shooting any pirate that crosses Liberty space, yes a BH can freely do it due to there being a bounty currently covering several targets within Liberty space. But only pertaining to those targets specifically listed on that bounty. The LPI does not need permission or a bounty in order to engage a pirate that is inside Liberty space.