For this specific case, it would be a grave mistake to license under anything, which does not contain a copyleft to make sure they release the source code as well and grant the 4 freedoms.
If I'm understanding this right, this app was written by the Australian government. Does the license for any libraries or other outside code they have included even matter?
Generally, governments have what is called "sovereign immunity" when it comes to civil lawsuits. They can only be sued if they decide to allow it. Some countries waive their sovereign immunity for specific laws.
For example US copyright law waives it for the Federal government, and so if the US government used your library without permission you could sue them for copyright infringement. It does not waive it in regard to the US states, however, and so if individual states used your library without permission you would probably not be able to do anything about it.
I have no idea if Australia has sovereign immunity from Australian copyright law. Google, Bing, and Duck Duck Go are all insisting on just returning results about the recent US Supreme Court case that said the waiver of sovereign immunity in US copyright just covers the Federal government, not the state government.
The Australian Govt respects copyright a dictacted by legislation, rather than some general immunity concept. So particular legislation (such as FOI, Archives Act, etc) may dictate that copies are made and kept. In general, the Govt pays fees as would a commercial entity, but many of the use cases could be covered by a legislative requirement or a fair use provision, and it is rather baroque, and the process of legal reform is quite slow.
As to software, I don't believe any part of the Australian Govt has been sued for violation of open source copyright, and it is generally taken quite seriously at agencies like CSIRO. It has always been a big talking point for MSFT and big integrators as a reason not to use open source though.
This is a pretty old-fashioned and US-centric perspective. The Wikipedia article explains that sovereign immunity never really worked like this in Australia, and the United Kingdom passed a Crown Proceedings Act to overcome it in 1947: https://en.wikipedia.org/wiki/Sovereign_immunity
In this case, a country that isn't the US trying to push a "pirated" version of an app against the will of the country that originally developed it would likely be hindered by DMCA takedowns directed at the app store operators.