Biggie, a staffordshire bull terrier cross, was shot dead in November.
The shooting caused much debate around the officer’s actions, but a police investigation found the constable was not in the wrong.
Lawyer Nicole Young said she and the dog’s owners Pat and Alena Wharram had not heard from police since December 2.
She said the matter was now with the CCC.
“We lodged a report of alleged serious misconduct on behalf of Pat and Alena Wharram and expect the CCC is in the process of assessing the report,” she said.
“We hope they will determine that an independent investigation is warranted.
“Our clients would still like to seek some compensation for the death of Biggie, but will await the outcome of our report to the CCC before lodging any such claim.”
In December, Mr Wharram was fined $400 by the City of Joondalup on accusations Biggie attacked another dog the day of the shooting.
Police were contacted for comment on the CCC lodgement, but declined.
In the weeks after the incident, the issue was further complicated when questions arose about the officer’s character. In a matter unrelated to the shooting, he tested positive for drug use and was suspended.
Police Commissioner Karl O’Callaghan said the man was drug tested after the shooting and cleared.