Can government regulators or law enforcement access a corporation's individuals-with-significant-control register?

Yes. One of the main purposes of the individuals-with-significant-control register requirement under the Business Corporations Act is to make this information available to government bodies and law enforcement carrying out legitimate regulatory, tax, or investigative functions, even though the register itself isn't a public document that anyone can search. A corporation generally has to produce the register on a proper request from an entitled regulator, tax authority, or law enforcement body, rather than treating it as confidential in the way it would treat, say, a competitor's inquiry.

This is really the point of the whole regime: transparency registers were introduced specifically to make it harder to use a corporation to obscure who actually controls it from the authorities responsible for enforcing tax, anti-money-laundering, and other laws, even while keeping that information out of general public view. So the register being private doesn't mean it's private from the government.

If your corporation receives a request from a regulator, tax authority, or law enforcement body for its ISC register, it's worth confirming the request is properly authorized, but corporations should generally expect to cooperate with legitimate requests of this kind rather than resist them.