Scott Morrison’s minority Coalition government is likely to face its first significant test on the floor of the lower house on Monday with a vote on whether to establish a national integrity commission.

Labor and the crossbench are preparing to push for the anti-corruption body in a motion moved by the Greens MP Adam Bandt after independent Kerryn Phelps is sworn-in as the member for Wentworth and the Coalition officially enters minority government.

A federal independent commission against corruption is supported by Labor and the entire crossbench – Cathy McGowan, Rebekha Sharkie, Andrew Wilkie, Bob Katter, Phelps and Bandt.

On Monday McGowan plans to introduce a private member’s bill to create an anti-corruption body, and a successful Senate motion calling for one will come to the House of Representatives for concurrence.

While an absolute majority of 76 is needed to suspend standing orders and force a vote on legislation, a government bid to shut down debate on the motion can be defeated with a simple majority, meaning Labor and the crossbench could defeat the government 75 votes to 74.

A push by Labor for the government to establish a national integrity commission on a bipartisan basis has stalled after the attorney general, Christian Porter, said in May that there was there was no “persuasive evidence” that current methods of tackling corruption were insufficient.

On Sunday, dozens of former judges heaped pressure on the Morrison government in an open letter, warning that existing anti-corruption agencies were insufficient and a new body was needed to improve trust in public institutions.

McGowan told Guardian Australia the attorney general Christian Porter had expressed concerns about the breadth of misconduct captured in the definition of “corruption” and that a national integrity commission should not unduly harm innocent people’s reputations or allow “vexatious” complaints.

She said those concerns had been addressed in her bill but she refused to “dumb it down”. The bill also contains reporting requirements for government agencies to show they have a plan to stamp out corruption and protections for whistleblowers.

After her win in the Wentworth byelection, Phelps joins the crossbench on Monday and will be able to vote and speak in the parliament following her maiden speech, scheduled for the same day.

Labor plans to meet the crossbench to find other common issues on which to agitate once Morrison no longer has the numbers to assert control over the House.

Phelps has said she wants to see action on Nauru, where at least 12 asylum seeker children and their families remain. The government has been quietly removing families from the detention centre to the mainland for medical treatment but Morrison has said they will be returned when their treatment concludes.

The government is also sweating on the potential for two of its MPs, Peter Dutton and Chris Crewther, to be referred to the high court over section 44 concerns. Katter has given the government some breathing room on that score, all but guaranteeing he would not support a referral, after Morrison committed funding to water projects in north Queensland.

Without Katter’s vote, Labor and the crossbench would need to convince at least two government MPs to vote with it to suspend standing orders to bring on any private member’s bill.

While Katter is not in favour of referring government MPs, and will not act to frustrate the government’s agenda, he has been positive about a federal integrity commission, bringing it closer to the needed numbers for debate.

McGowan, the member for Indi, said the coming fortnight was shaping up as one of the most important sittings of the year.

“From my perspective, the integrity commission is one of the most important issues,” she said. “The government needs to demonstrate why it wouldn’t support it, or put forward some very compelling arguments on why it doesn’t think it needs to happen.

“The other important issues we’ll be seeking to bring forward are the New Zealand offer [to accept 150 asylum seekers from Nauru], climate change and what we are doing about it, and of course there is the issue of the referrals. We’ll be looking at all of it.”