My favorite "Jason is petty" fanfic is
this one: "Red Hood: Lost Days AU. After finding out that he remains unavenged, Jason returns to Gotham, but instead of deciding to kill Bruce, he decides to channel his rage into a more creative endeavor: spray-painting Gotham until the message is loud and clear to everyone: f*ck Batman."
And here is
another fanfic I like; Jason runs away, doesn't search for his mother and becomes a soldier instead.
Jason as policeman could be really fun. I'd like to read an Elseworlds' story without superheroes in which Jason is a policeman who tries to fight corruption at day, and a crime lord who tries to control crime at night.