The Bronwyn Bishop helicopter affair has morphed from a big personal embarrassment to a serious issue for the Abbott government.

Now Labor has referred her A$5227 charter to the Australian Federal Police, the speaker finds herself in a compromised position while they decide whether to launch a formal investigation.

It is not clear how long the AFP will take to determine if there are grounds to move to that next stage, but one supposes they are likely to have done so by parliament’s August 10 resumption. If there was a formal investigation, Bishop would presumably, at the very least, have to stand aside.

In the meantime, her spokesman says that “the speaker does not intend to make any further comments at this stage, whilst the AFP are considering a referral to them on this issue”.

The paperwork covering charters for parliamentary presiding officers is very explicit about the circumstances in which a charter can be used (“official purposes”) and reminds signatories that it is a serious offence under criminal law to give false or misleading information.

So far, Bishop has not released the form that she signed. She has been reluctant to say much at all, apart from that she understood the travel was within the rules but she was repaying the money (onto which is added a 25% penalty).

If Bishop’s claim was “within the rules”, surely it would have to include something we don’t know. A Liberal fundraiser for a state election candidate hardly sounds like an occasion relating to the speaker’s “office holder duties”.

Beyond the question of whether the claim was inside or way beyond the rules is the matter of how it looks to be spending such a huge amount on what is little further than a suburban commute – a journey of around an hour each way by car.

Can a government that bangs on about ending “the age of entitlement” have as its speaker someone willing to waste thousands of dollars of public money like this? Can a Coalition that is always looking for and condemning any rorting of welfare payments allow such profligacy by a senior parliamentary office holder? The funds for welfare recipients and parliamentarians’ entitlements all come from the same source – the ordinary taxpayer.

Often in politics it’s the little things that really resonate – the barbecue talking points that reinforce stereotypes. Bishop’s action puts the whole government on the nose in a community that is angry and cynical about politicians. It reflects extreme arrogance, and an attitude that is completely out of touch.

It’s extraordinary almost beyond belief. Who would do this? What could she have been thinking? If she had a complete brain snap and said “get the helicopter” instead of “book the Commonwealth car” surely she or her staffer would have thought afterwards that this madness had better be paid for personally and quickly.

Bishop has been a divisive and poor speaker. Admittedly it is a hard job, especially in the sort of feral House that we have. But she doesn’t try to be objective.

This, together with the Coalition’s pursuit of former speaker Peter Slipper, means Labor will go as hard as it can against her.

Tony Abbott would be foolish to waste political capital on trying to defend her. Joe Hockey didn’t even bother to this week. The government would look bad if she were found to have a case to answer under the rules. Even if the police say there is not going to be a formal investigation, Abbott would need a lot of political capital to back an office holder who behaves like this. And he doesn’t have that sort of capital.

And if Bishop and the government dig in, there will be plenty of other likely extravagances for the opposition and the media to search for. She spent more than $300,000 on overseas trips in a year. A few luxuries there, one would think.

The speaker, once having been elected by the House, can in formal terms only be removed by the House. Not that one can see Bishop, if the government decided to cut her loose, forcing it to move a motion to get rid of her.

But if Bishop, who has thrown out some 400 MPs for bad behaviour, had to vacate the chair for doing the wrong thing, it would be more than a little ironic.

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