It was the Governor-General not the monarch (Edward VII) who refused to sign the bill, on orders from the British Parliament, that would have restricted the right of Australians to appeal to the British Privy Council. As the British Parliament still held some control over decisions that affected citizens of the empire the British Parliament had the right to tell the GG not to agree to the legislation (he resigned over the issue).I know but apparently an Australian bill was refused sovereign assent in 1906, can't remember what it was called.
At that time Australia only had control of our internal affairs and anything that affected the rights of the subjects of the empire had to be approved in London, which is why the GG took the action he did.
It wasn't the monarch that gave those instructions but the PM and his government in Britain.
We actually maintained the right to appeal to the Privy Council until 1984. It was one of the last connections between Australia and the UK in any real sense. Now it is a shared Head of State and nothing more.