Mr Byrne is facing growing pressure, with both rank and file officers and civilian staff considering confidence votes in his leadership.
Last week insisted he would not resign following a marathon session of his oversight body, the Policing Board.
The latest controversy to hit the force erupted last week when High Court judge Mr Justice Scoffield ruled that two probationary PSNI officers were unlawfully disciplined for an arrest made at a commemoration in 2021 on the Ormeau Road in Belfast.
The event marked the anniversary of the 1992 Sean Graham bookmakers attack, in which five people were murdered by loyalists.
The judge said the two officers had been disciplined to allay a threat that Sinn Fein could withdraw its support for policing - something the party has firmly denied.
Mr Byrne initially said he accepted the judgment, but on Thursday indicated that an appeal was being considered and said it was inappropriate to make any further comment.
The chair of the Police Federation for Northern Ireland Liam Kelly last week expressed “disbelief and anger” at the statement.
Speaking to his party members at the party's summer barbecue on Saturday, TUV leader Jim Allister insisted once again that Mr Byrne must go.
“This week the High Court shone a long overdue spotlight into the dark corners of the PSNI, confirming what many believed, that Sinn Fein is the puppet master of policing?" Mr Allister told party members.
“Despite all their twisting and dissembling, it is clear that Sinn Fein demanded the sacrificing of the two young officers and our pathetic and political Chief Constable delivered.
"The subsequent pretence by lying Sinn Fein that they would never withdraw support for policing is exposed by what their Chairman, Kearney, wrote at the time in An Phoblacht: 'Our engagement with the PSNI and the police accountability mechanisms, such as the Policing Board, has been consistent and constructive. But it has also been conditional ... some have clearly made an assumption that Sinn Fein's leadership can be taken for granted, that's a mistake.'
“The message is clear and fits precisely with them threatening jelly-kneed Byrne.
“It is not enough that Simon Bryne goes, there must be root and branch change, focused most particularly on expunging from the processes political diktat. A good starting point would be to import into the Policing Board arrangement the same ban as we secured for Special Advisers, namely, that no one with a serious criminal conviction can hold such office.
“That would be a demonstrable indication of real change and rid us of the obscenity of a convicted terrorist sitting on the Board and chairing one of its committees.
“The days of pandering to Sinn Fein need to be over if Unionist confidence is to be restored to the PSNI.”
The PSNI, Policing Board and Sinn Fein were all invited to respond to Mr Allister’s comments.
The Police Federation has called an extraordinary meeting of its executive central committee this Wednesday when it said a confidence vote may take place into the PSNI leadership.
The Superintendents’ Association of Northern Ireland, which represents more senior officers, said it was standing with the federation.
In a social media post it said the “senior executive team must listen to our collective concerns”.
Nipsa, which represents a number of civilian police staff, is also set to hold an extraordinary departmental committee meeting of police staff representatives next week, during which it will be assessed if there is a demand for a confidence vote in Mr Byrne.
Ulster Unionist leader Doug Beattie has called for Mr Byrne and his deputy Mark Hamilton to resign.
The court ruling heaped further pressure on Mr Byrne, who was already facing questions about his future after a major data blunder led to personal details of officers entering the public domain and getting into the hands of dissident republicans.