The Governor General has in recent times prorogued Parliament when the Prime Minister asked them to. Ie. "This is politically nasty. Let's hit the pause button and come back when things are better and we're not about to be ejected from power..." And that has been politically controversial. Historically the Governor General just says yes because they want to avoid playing a political role at all (ie. preserving this convention that the Monarchy is really just a decoration of our government).
In practice Taylor Swift might have more.
If he does something that is openly obviously defying the will of the democratic majority, sure. But there are a lot of ways he can put his thumb on the scale in more subtle situations where the legal course of action is unclear.
For example, if the Queen had refused Boris Johnson's request to prorogue parliament in 2019 -- which was later ruled illegal anyway -- I doubt it would have led to the end of the monarchy.
Constitutionally, the king of Canada is the commander in chief of its armed forces, provides consent or assent to all laws passed by Parliament, has some immunity from prosecution, and has a pardon power. In actual practice, most of those powers are performed perfunctorily by delegates based on either action by Parliament or by recommendation of ministers determined by the Prime Minister.
https://en.m.wikipedia.org/wiki/1975_Australian_constitution...
In principle this power still exists. Whether Charles could pull off the same trick depends on the political situation on the ground.
I'm old enough to remember it and remember a statement from the palace saying something like "The Queen is watching events in Australia with interest" but I don't think she took an active part.
I quick search reveals this. I don't know this site but if true then some letters seem to confirm the above. She told the GG to obey the Australian Constitution.
https://constitution-unit.com/2020/07/16/palace-letters-show...
https://en.wikipedia.org/wiki/Alleged_CIA_involvement_in_the...
It's true that the palace was involved in discussions with the GG to a greater extent than most Australians though was acceptable though.
However this was before the 1986 Australia Act which cleared up the ambiguity around that.