Samantha Lee is said to have failed to make "the correct investigative inquiries" when Couzens exposed himself at a fast-food restaurant in Kent.
At the misconduct hearing, she was also accused of lying about her actions.
Ms Lee, who is no longer a police officer, denies gross misconduct.
She also denies breaching the force's standards.
The misconduct allegations relate to how Ms Lee investigated two instances of Couzens exposing himself to female members of staff at the McDonald's drive-through restaurant in Swanley, Kent, on 14 and 27 February 2020.
The police disciplinary hearing was told that Ms Lee visited the restaurant on 3 March, interviewing the branch's manager Sam Taylor, hours before Ms Everard was kidnapped by Couzens in Clapham, south-west London.
Paul Ozin KC, for the Met Police, said after the incidents were reported to the force on 28 February, the matter was recorded as "less urgent" than other matters the force had to deal with.
He said a computer check, which was made after the restaurant manager made the call, confirmed the black Seat Exeo had been registered to Couzens since January 2018.
Mr Ozin added: "There is no standard check that takes place to see whether a suspect in criminal police cases are police officers."
The hearing was also told about claims that Ms Lee lied about her knowledge of the restaurant's CCTV.
Mr Ozin said Mr Taylor claimed to have explained to her that while the drive-through CCTV soon deleted automatically, other CCTV footage - which he showed her - depicted Couzens' car.
Mr Taylor also showed her receipts that showed the last four digits of Couzens' card on both occasions, the hearing was told, as well as witness statements taken from two members of staff.
However, Mr Ozin told the hearing Ms Lee says she was not shown any CCTV footage but was instead shown how the CCTV system worked.
He said she had accepted she took possession of receipts and witness statements from Mr Taylor.
In a report made after attending the restaurant, Ms Lee recommended Couzens be arrested and questioned.
Mr Ozin said Ms Lee said she believed this report would be allocated to a different team to follow up on.
However, Mr Ozin said she did not put the witness statements and the receipts in a sealed evidence bag, instead keeping them in a pocket in her body armour.
Mr Ozin said: "One of the central issues of this case is whether there has been some horrible misunderstanding.
"It [the evidence] is supportive of the unpalatable conclusion that PC Lee just did not bother to get the CCTV, even though she knew it was important, instead relying on others to do that instead of her, and that she later lied to others when she knew that the stakes had escalated astronomically."
If Ms Lee is found to have committed gross misconduct, she could be banned from serving as a police officer again.
In March, Couzens was sentenced to 19 months in prison after admitting three counts of indecent exposure. He was already serving a whole-life term for the kidnap, rape and murder of Ms Everard.
The misconduct hearing is expected to last seven days.