The head of the Catholic Education Commission of Victoria has expressed regret that Simon Birmingham lost his portfolio over schools funding reforms, appearing to take credit for ousting the education minister.

Stephen Elder made the comments to the Australian Financial Review after he headed its education power list, a victory lap capping the announcement the sector had extracted $4.6bn more over 10 years for Catholic and independent schools after Scott Morrison ousted Malcolm Turnbull and replaced Birmingham with Dan Tehan.

It comes as the Grattan Institute’s school education program director, Peter Goss, released analysis that Catholic schools could protect low-income parents from fee rises at less than a tenth of the cost of the $1.2bn school choice and affordability fund.

Elder – a former state Liberal MP in Victoria – reportedly agreed that he had helped unseat Birmingham, claiming that “always concerns me because I didn’t want to have a part in that”.

He said that in January the Victorian Liberal president, Michael Kroger, had arranged for him to meet Tehan, who was social services minister.

He said Birmingham “wasn’t listening” to the sector’s concerns with the Turnbull government’s 2017 schools funding changes and Kroger was concerned the issue was hurting the party in the polls.

Elder said that 750,000 Catholic school students nationwide were likely to have to pay higher fees and some schools faced closure, which required “some drastic action” from the Catholic Education Commission including making 280,000 robo calls into marginal Liberal seats in Victoria.

“We had no option but to go to the public square and state our case,” Elder reportedly said. “I’m not responsible for anything else that happens after that.”

Elder said that funding schools on the basis of the socioeconomic status of parents had harmed primary schools in higher SES areas such as Holy Rosary in Kensington, Victoria, which had a lower SES score than Geelong Grammar.

At the time the federal government was waiting for the Chaney review by the National Schools Resourcing Board which in July recommended using a direct measure of parents’ income to calculate schools communities’ capacity to pay.

That recommendation became the basis of $3.2bn of the package announced by Morrison and Tehan on Thursday.

Elder thanked Birmingham – who became trade minister in the reshuffle and is still the deputy leader of the government in the Senate – for “his contribution to public life and the sacrifice that entails”.

“I hope [Senator Birmingham] learned from this you need a multitude of counsellors,” he reportedly said. “He needed to seek advice more broadly and I am sure he would have made much better public policy decision.”

Asked if he regretted what happened to Birmingham, Elder reportedly replied, “Yes … I mean, to be publicly humiliated was not what I wanted.”

Negotiations between the Morrison government and the Catholic and independent sectors helped secure a $1.2bn “choice and affordability fund” on top of the $3.2bn to change the schools funding formula.

Over a decade $718m will go to Catholic schools and $485m to independent schools – based on the number of schools and students in each system – although criteria for the fund have yet to be developed and agreed.

The fund has been derided by Labor and education experts as a “slush fund” and the Greens have suggested they could move to disallow the regulation establishing the fund in the Senate.

Goss told Guardian Australia that under successive education reviews conducted by David Gonski “all schools are equal” but “the choice and affordability fund means that some are more equal than others. It shouldn’t be there.”

Goss said only a “very small number of Catholic primary schools” faced substantial fee rises and “the typical family” in schools with high SES score could be expected to afford increases to $8,000, which is still less than the cost of educating primary school children ($11,ooo).

Goss said there was “some merit” in helping families achieve “affordable choice” but that did not mean the $1.2bn fund should be spent making those Catholic schools “low fee for everyone”.

“If the typical family can afford $8,000 fees, then setting them at $2,000 is a windfall gain for most of them,” he said.

Goss said of the 10,000 students in the 40 schools likely to be most affected, MySchool data shows that just 125 students are in the bottom two SES quartiles.

The government could pay $4m a year to subsidise their fees down to $2,000, which he said is “an order of magnitude” less than the $70m a year set aside for Catholic schools in the choice and affordability fund.