I think a good aspect to think about is that such an operation would require a tremendous amount of intelligence signals and traffic, which would be intercepted and made use of through MAGIC/PURPLE. Granted, that early in the war the codes weren't 'fully' cracked. Having said that...its unlikely if not impossible the Allies wouldn't have some idea as to an invasion of Australia.
But....and here is something I just had popped into my head, and so requires a bit of handwavium, we have scenario wherein the Japanese know the codes are broken, and pull almost a reverse-Midway by making lots of noise about invading Australia but in fact throwing off the allies. But even thats is incredibly unlikely.