The modern facilities are a step up from the local public high school, but it's not King's School or Melbourne Grammar. There's no swimming pool or equestrian centre. Most of the students are working and middle-class kids whose families want them to get a good Catholic education. 

For Tasmanian senator Jacqui Lambie, it will always be a special place.

As she explained in an emotional speech to the Senate in March, Lambie found herself at the "bottom of the crap pile" when she was medically discharged from the army in 2000. A single mum on the disability support pension, Lambie couldn't afford the fees, but St Brendan-Shaw allowed her sons to attend free of charge for three years.

It's something she's still grateful for.

So when Education Minister Simon Birmingham was courting Lambie to support his Gonski 2.0 school funding model, she knew where to take him. With a Senate vote looming, Birmingham flew to Tasmania this month and spent half a day with Lambie at St Brendan-Shaw talking education policy.

It was to prove a trip worth making.

Wedded to the "Gonski 1.0" model it devised when in office, Labor was implacably opposed to the government's changes. Pure politics made a deal with the Greens look shaky. Small-government senators David Leyonhjelm​ and Cory Bernardi​ wouldn't vote for extra spending on schools. 

This meant Birmingham would have to win over the other 10 members of the Senate crossbench – including Lambie. There was no room for error and everything was on the line. The government would head into the six-week winter parliamentary break on a euphoric high or a crushing low.

Securing Lambie's vote wouldn't be easy. With a short fuse and an aversion to spending restraint, she almost never votes with the government. An analysis conducted earlier this year shows she has sided with the Coalition just 30 per cent of the time on substantive votes in this Parliament. 

Lambie was also worried about Catholic schools. Under Birmingham's new model, funding for Catholic schools would keep growing but less generously than under the current legislation. With reduced funding, Lambie wondered, would schools like St Brendan-Shaw still be able to help out struggling families?

Birmingham left Tasmania hopeful, but not assured. Lambie hadn't committed to voting yes, neither had she ruled it out. He was still in the game.

Six weeks earlier, an alert had gone out from the Prime Minister's office. The PM was to hold a surprise press conference with Birmingham in Sydney. As journalists arrived, they noticed there were three lecterns. Who else, they wondered, would appear?

When Turnbull and Birmingham strode out alongside David Gonski, Australia's school funding debate turned on its head.

The respected businessman had conducted the landmark review into school funding for Julia Gillard in 2011. His name had become shorthand for Labor's big-spending funding deals with the states. Now here he was, standing alongside a Liberal PM and education minister endorsing their policies.

It was a stunning moment that had been months in the making. Liberal insiders still can't believe they pulled it off without it leaking. 

When he became education minister in 2015, Birmingham knew he faced a wicked political and policy dilemma. He would never get Tony Abbott's 2014 policy, which would reduce school funding by $30 billion over a decade, through the Senate. But he would never be given enough money to match Labor's school funding promises.

So he had to find a way through. Soon after taking the job, with the help of then-NSW education minister Adrian Piccoli, he reached out to Gonski for advice. And he continued to consult with him over coming months as he developed his plan for a new school-funding model.

Birmingham believed he could get Gonski's approval for a plan that delivered less funding than Labor but distributed it in a fairer way. 

Turnbull, friends with Gonski since childhood, had even bigger ideas. He believed he could convince the media-shy businessman to go public in a way that could capture people's attention. 

"Without his endorsement, this would never have got off the ground," a senior Liberal source says. "He gave us a feather to fly with."

At the press conference, Turnbull boldly declared he would "bring the school funding wars" to an end with his $18.6 billion spending plan.

He was far too optimistic.

The Catholic education sector quickly unleashed a fierce campaign against the government, with warnings of parish schools closing and fees soaring by $5000 a year.

Birmingham privately admits he didn't see the backlash coming. He knew the Catholics wouldn't be happy to lose their prized funding arrangements, but the ferocity of their response caught him by surprise.

In the days after the announcement, Coalition MPs' inboxes soon started filling with emails from concerned Catholic parents. By providing his colleagues with school-by-school figures for their electorates that showed funding going up, Birmingham headed off a party room revolt. A landmine, though, was threatening to explode.

After eight years in Parliament, low-profile WA Liberal senator Chris Back announced his retirement last week. A former member of the WA Catholic Education Commission, Back had told Birmingham privately he couldn't vote for the legislation as it stood. On Monday he went public with a threat to cross the floor.

All Birmingham's work to sway the crossbench, including agreeing to $5 billion in extra spending, was at risk of coming to nothing.

In the end, the promise of a one-year reprieve before Catholic schools lost their cherished systemic funding agreements locked in Back's vote.

It looked dangerously close to the "special deals" Birmingham had railed against but, at a cost of just $50 million, it was worth it. The concession also helped get Lambie, the 10th and final vote required, over the line. Thanking Birmingham for taking the time to visit her state, she reluctantly announced: "I am swayed by what he's doing." 

And she held firm, despite the efforts of Catholic education lobbyists who camped outside her office on Thursday in a bid to change her mind. After squeaking through the Senate a few hours earlier, the amended bill passed through the House of Representatives at 2.01am on Friday.

Brinkmanship is vital for the Turnbull government. Each of its major legislative wins – the Australian Building and Construction Commission, the company tax cuts and now school funding – has had to survive the turbulent guard of the Senate crossbench, with mixed results.

Recall, for instance, the watered-down ABCC included a two-year transition period demanded by Victorian senator Derryn Hinch in exchange for his support. Months later, he backflipped, having undertaken a listening tour over the summer, and helped the government restore some of the commission's teeth.

Hinch was also at the centre of comical manoeuvring over the so-called backpacker tax, oscillating between four different positions in four days, staging rambling press conferences in the centre of Parliament, before the government went around the crossbench and reached a deal with the Greens for a 15 per cent tax.

Despite the shambolic process, in each case the government has ended up getting pretty much what it wanted. It has had to compromise, certainly – most notably on its enterprise tax plan, where the Senate would only accept tax cuts for companies with turnover less than $50 million.

"On any view of the world, the Senate has largely passed the government's reforms with the corners rubbed off – and it was ever thus," says John Daley, executive director of the Grattan Institute, who strongly backs the revised Gonski 2.0 package, and views the crossbench's amendments as an improvement.

The national resourcing body to oversee school funding "makes a lot of sense", Daley says, given the historical fights over school funding and bickering between the Commonwealth and the states. He also gives the shorter implementation time frame a tick. "Bringing the schools at the bottom up faster is a good thing," he says. 

More broadly, Daley argues this week's success undermines the claim that governments' agendas are being hamstrung by the Senate. That idea has been pushed by former prime minister Tony Abbott, who earlier this year warned the Senate had become "a house of rejection" rather than a house of review, and was delivering "gridlock, not government". 

And in a well-publicised speech in February, chairman and former Treasury boss Ken Henry lamented the state of politics, arguing it was mired in "dysfunction", and fuelled by populism and tribal tensions within parties. While his criticism was chiefly directed at a failure to tackle big-picture economic and taxation reform, he also took aim at the "shambles" of climate change and energy policy.

Daley, also frustrated by the breakdown of climate policy in recent years, says the Turnbull government's energy plans "by and large look pretty sensible". This week the PM confirmed he would proceed with parts of Chief Scientist Alan Finkel's​ review, including controls on gas exports, and abolishing a merits review process by which transmission companies have gouged consumers for extra cash.

A proposed Clean Energy Target, however, remains more contentious, with members of the Coalition party room already publicly positioning against it. As the next cab off the rank, expect that issue to bubble away over the winter break and into the second half of the year.

In Daley's view, the government's steady, orderly pace to reform is working, even when subject to compromise.

"They haven't picked off too much at once. They're only trying to land three really significant reforms at the moment," he says, naming school funding, higher education and energy. "And they're happy to sequence them. One of the things I think we have learnt – or re-learnt – is you cannot get too much through at once. You run out of bandwidth. That's certainly a criticism that was levelled at the Rudd government."

Celebrating the Gonski win on Friday, a beaming, fist-pumping Malcolm Turnbull declared he had landed "the biggest reform in Commonwealth school funding ever", and the school funding wars "should now be over". He conceded the extra $5 billion in expenditure – $1.5 billion over the forward estimates – "wasn't what we planned to do", but acknowledged getting out the chequebook was often "what you need to do to get legislation passed".

"The alternative is that you don't get anything done," Turnbull observed, taking the opportunity to once again note the government had exceeded the expectations of those who believed it was "in office but not in power".

One government frontbencher with a trained political eye believes this week's deal is a big win, though not necessarily a game-changer. "We had a legacy problem where we made a major mistake," he says, namely the horror 2014 Abbott budget which defined the Coalition's position on school funding as one of mammoth cuts.

Now, nervous marginal seat MPs can stride into schools in their electorates and put a dollar figure on the amount of new money they will get under Turnbull. Even Abbott, who complained in party room meetings about the affordability of extra expenditure on schools, was worried about Catholic schools in his Broken Bay diocese losing out.

Unlike the construction watchdog or the company tax cuts, Gonski 2.0 could be a barbecue stopper. "Education funding doesn't matter to parents, schools do," the frontbencher observes. "This is why the marginal seat members, even the Catholics, [are saying] they're happy with this, because it gives them something to talk about."

That doesn't mean he believes Malcolm Turnbull will catch much of a break in the polling stakes.

"I don't think think the polls will change. Any politician who is looking for an applauding electorate is in the wrong game," he says. "Your achievements are your banked credibility for your next series of promises. And we have a much better story to tell than we did two months ago."