Horrific sights in Australia right now!
As for the scary snakes that can easily kill you that Australia is famous for - yes, they are probably feasting right now, but I remember watching news on a prior mouse outbreak when I was a kid, and it showed what happened to the snakes during the cold months - the rodents find them while they are hibernating and eat them! Crazy!
I think that they should just release a bunch of coconut crabs! Fully armoured! Nah.. you don't want another invasive species.
Sounds like they need to make a call to Joseph Carter - the Mink Man. NSFW tags for rats getting killed - not for the squeamish. He uses mink and dog teams to take out rat infestations. He is having some pretty good success right now training an Argus monitor lizard right as well, and they are native to the region that this infestation is happening. He is working with raptors as well. Pretty interesting videos, but it does show rats being killed, so NSFW.
Unfortunately, poison is the only realistic answer here. Sucks, as it is going to take out a lot of other wildlife out, but there is little other recourse. This happens in Australia regularly (as well as the rabbits!). Nature is full of cyclical events like this when some species explode in number, eat everything up, and then die naturally, and you will see similar population booms and busts with predators as well mirroring these explosions.
I am not afraid of mice, but being there would probably make me develop a phobia of them, no doubt!
PETA does some good work for things, but I swear they mostly do stupid things that makes the majority of people who love animals, to hate them. Catch and release my butt. Poison and fire is the only way for now.
I think it is time that science starts investing a bit more on how to eradicate these pests a bit more effectively (and hopefully poison-free!). I also think people need to look at better strategies of grain storage (if for nothing else than to make sure that I don't eat bread made with grain and mice pee and poop!!).