Labor has renewed its attack on Peter Dutton’s eligibility to sit in parliament, citing an agreement to hire a special needs teacher as the missing piece to make a case against the home affairs minister’s interests in childcare centres.

The shadow attorney general Mark Dreyfus said on Thursday it was “extremely significant” that Dutton continues to sit in parliament while the government has a one-seat majority, arguing that he is “quite probably” ineligible.

Although the solicitor general Stephen Donaghue provided advice that Dutton is likely “not incapable” of sitting, Dreyfus said the advice “made it clear that the matter remains in doubt”.

Constitutional law expert Anne Twomey has noted that the advice reveals that payments had been made by the commonwealth to RHT Investments – of which Dutton and his wife are beneficiaries – under the inclusion support program.

The advice states the Camelia Avenue childcare centre, operated by RHT Investments, has received a $15,640 subsidy to hire a special needs teacher.

Section 44(v) of the constitution bans parliamentarians from having a “direct or indirect pecuniary interest in an agreement with the commonwealth”.

Donaghue concluded that participation in the program was likely to constitute an agreement with the commonwealth, but said the “better view” is that Dutton does not have an interest in it because the funding was given for specific purposes to hire a special needs teacher.

“Assuming the Camelia Avenue childcare centre complied with that condition, it therefore appears unlikely that the funding would have generated a surplus for RHT Investments that could have been distributed to Mr Dutton,” he said.

Dreyfus told reporters that it is “now clear … that Peter Dutton must be referred to the high court”.

“This further document that's come to light makes it clear that there is an agreement between Mr Dutton's childcare centres and the commonwealth and that's why … we think it's a clear breach of section 44 of the constitution.”

Dreyfus said the constitutional prohibition is “absolutely clear” regardless of “the question of what the purpose of the commonwealth funds is”.

For the high court to hear Dutton’s case before the next election, the House of Representatives would need to refer the matter.

On 23 August the House voted 69 votes to 68 not to refer Dutton to the court, despite four crossbench MPs Adam Bandt, Rebekha Sharkie, Cathy McGowan and Andrew Wilkie voting with Labor to refer.

With Malcolm Turnbull’s resignation and the vacancy in the seat of Wentworth, numbers will be tight for a further push to refer Dutton. The Liberals and Nationals still have a slim majority of 75 of 149 MPs, suggesting that Liberal MPs would need to abstain to help Labor get the numbers to refer Dutton.