The finance department, asked by the Australian Federal Police (AFP) to investigate Bronwyn Bishop’s A$5227 helicopter ride between Melbourne and Geelong, now finds itself sitting on a very hot seat.

If the department decides Bishop breached the entitlement rules, the question becomes whether this is referred back to the AFP for formal investigation or is accepted as a mistake.

If, however, finance concludes Bishop was within the Speaker’s entitlement when she claimed for attending a state election fundraiser, the government will be under heavy pressure to amend the rules which, it seems, are wide and woolly.

The opposition has questioned the AFP’s decision to refer the issue on, arguing that its refusal to investigate the allegations against Bishop would be inconsistent with the approach taken to allegations against one of her predecessors, Peter Slipper.

The police are anxious not to be seen to be avoiding their duty. A spokesperson said the AFP had “not refused to investigate” but had sent the matter to finance for assessment in accordance with the so-called Minchin protocol on dealing with alleged breaches of entitlements.

This protocol was approved by then-special minister of state Nick Minchin in 1998.

Under it, when an allegation of misuse is made, the finance department investigates. If the matter is minor, the MP gets a please explain.

If more serious, the matter goes to a high-level departmental committee chaired by the department’s secretary (currently Jane Halton). This committee may or may not seek an explanation from the parliamentarian.

If it considers the matter possibly criminal, or is unclear about questions of law, the committee will obtain advice from the secretary of the attorney-general’s department as to whether it warrants referral to the AFP.

The finance secretary makes the final decision on a referral, but in practice this would be on the advice of the attorney-general’s department secretary.

The Minchin protocol is supposed to allow politicians to fix up mistakes, as well as to put the entitlements process at arm’s length from the politicians.

But the difficulties of asking bureaucrats to rule on their political masters are obvious. It would be better if the initial investigations were done by some independent tribunal.

Also, if the rules are found to cover such a claim as Bishop’s (forget that she’s repaying, which would not have happened without the fuss) then they should be tightened. But any workable rules will always be, to an extent, hostage to the integrity of those making the claims.

The AFP has addressed the suggested difference in treatment in the Slipper issue by saying that it received a referral “along with associated information” to investigate fraud allegations. It said this referral did not come from the finance department. “Material relating to this matter was sought from finance, which provided it to the AFP.”

Bishop – who has asked that two other charters she ordered also be considered (not helicopters) – continues to insist her claim was within the rules, essentially daring anyone to prove that it was not.

“I was a guest speaker. I find it unconscionable for you to think there are certain people in the Australian community that are not entitled to hear from members of parliament,” Bishop said defiantly.

“I’m speaking about the parliament and how it works and that’s what I did.” (No recording of what she spoke about has surfaced.)

Meanwhile Bishop subtly rearranges her story. “Whilst my understanding is that this travel was conducted within the rules, to avoid any doubt, I will reimburse the full costs,” she said last week. Now she is saying: “it was done within the entitlement. When I saw that figure, I saw it was too large and that’s why I’m repaying it.”

Bishop has not indicated any other activities around this event which were more “official”. Yet a while ago Tony Abbott explained his lateness to a partyroom meeting by saying he had attended a function that morning to justify claiming a trip to Melbourne for a party fundraiser the night before.

Abbott, who has had to repay claims in the past, is standing behind the embattled Speaker. “Look, the important thing is that Bronwyn has admitted that it was probably an error of judgement and she has repaid the money,” he said on Saturday.

But Bill Shorten, no doubt crossing fingers that the Liberals don’t discover any embarrassing Labor claims, is continuing to escalate the attack. “Does Mr Abbott have the character to tell Bronwyn Bishop to stand aside?” Shorten said on Sunday. “Because he knows and the world knows that this arrogant misuse of taxpayer funds needs to be held to account.”

Bishop and Abbott are gambling that the rules on entitlements will be too broad for the Speaker to be declared in breach, and that the news cycle will move on after a couple more days.

On the other hand, some see the helicopter ride as one of those lightning rods that will continue to seriously outrage the public, a less dramatic version of the knighthood for Prince Philip.

Listen to the newest Politics with Michelle Grattan podcast, with French Ambassador Christophe Lecourtier, here.