The federal government’s moves to avoid an election battle with the Catholic education sector is threatening to reignite the funding wars on another front, with the Australia Education Union launching a marginal seat campaign warning it against doing “special deals with private schools”.

On Monday the AEU – the peak union for public school teachers – began a publicity blitz in 18 marginal seats, claiming the government has cut funding to public schools while seeking to appease the Catholic sector.

The campaign will target Peter Dutton’s seat of Dickson in Queensland, Chisholm in Victoria – where sitting MP Julia Banks has announced she will quit because of “bullying” within the party – and marginal Nationals seats such as Page in New South Wales.

The AEU says it will spend about $200,000 launching billboards calling for “fair school funding” on highways and in shopping centres, as well as mobilising volunteers for a door-knocking campaign in NSW and Victoria.

In all of the seats – but particularly in Queensland – volunteers will hold “personal conversations” with voters outside public school gates, according to campaign information seen by ​Guardian Australia.

“If Prime Minister Scott Morrison thinks the solution to ending the school funding wars is by doing a special deal with private schools while ignoring public schools, then he is very much mistaken,” the head of the AEU, Correna Haythorpe, said.

“More than 2.5 million students attend public schools across Australia. They have been forgotten by the Morrison government.

“Our week of action will be a big reminder that these children deserve the same treatment, the same school funding, and the same chance at achieving their potential as do private school students.”

The Catholic sector is vocally opposed to the government’s Gonski 2.0 funding arrangements because it removes the system weighted average, which allows it to keep fees low in schools in high-income areas.

The sector has waged a fierce campaign against the Coalition policy, writing to parents before the Batman and Longman byelections to warn of possible fee rises and school closures. Labor has promised to restore $250m to the Catholic sector in the first two years if elected.

The possibility of a Catholic sector campaign against the government at the next election has spooked backbenchers and Morrison has given the new education minister, Dan Tehan, a remit to solve the issue.

But Tehan faces a delicate problem. The independent school sector has signalled doing a “special deal” with the Catholic sector would risk reigniting the so-called funding wars. Guardian Australia has previously reported an agreement with the Catholic sector could prompt a backlash from the states.

While the states signed up to an interim Gonski funding agreement last year, they are yet to sign up to a long-term funding deal.

Many of the states were unhappy with the funding arrangement because it required them to significantly ramp up public school contributions, while advocates within the sector such as the AEU are concerned the policy cuts funding growth relative to the previous Labor government’s arrangements.

“Our week of action will be delivering a clear message: our local public schools are missing out on billions in urgently-needed funding because of school funding cuts by the Morrison government,” Haythorpe said.

“Community support is firmly behind funding public schools and behind the fair funding now campaign. Recent polling showed that three in four residents said public school funding is personally important to them as an electoral issue.”

“Voters, parents and families tell us that they are very happy with their local schools, teachers, principals and support staff, and do not want them to be savaged by Scott Morrison’s cuts to school funding”

Guardian Australia has contacted Tehan’s office for comment.