Reid is serving a 23-year sentence for raping and killing deaf woman Emma Agnew in 2007, and the rape, attempted murder and robbery of a 21-year-old student nine days later.
Murray was sentenced to 50 hours' community work for that and was back in trouble early on the morning of Sunday December 8 last year.
Statements from police officers, viewed by APNZ, described how Murray was seen by police travelling at 64 kmh in a 50 kmh zone in Mechanics Bay, Auckland.
A following police car activated its lights and sirens but Murray sped up to 87 kmh and travelled some distance before stopping. She told Constable Dan Whalen she thought he was trying to get past her, so she accelerated and did pull over.
When Mr Whalen tried to give her a court summons, he said Murray told him she wanted to be given her rights in Maori.
Then, in what was described as an "agitated state'', she said Mr Whalen lied to her and misinformed her.
"She made the threat that she was going to take me down and me look stupid in court,'' Mr Whalen said.
Senior Constable Tipi Poi said Murray was saying she was being harassed and her arrest was unlawful.
When told she was not under arrest, she "argued the point''.
Ms Murray told APNZ she wanted a re-hearing into the failing to stop charge, but was yet to hear back from the court. She said she planned to defend the charge but had instead been found guilty at a hearing she wasn't aware of.
Ms Murray has asked police for CCTV footage that she said would show that she had stopped twice on the morning in question.
She was also unhappy she was fined, saying that highlighted sentencing inconsistencies in the courts after the judge in the smuggling prosecution imposed community work because Murray didn't have financial means.