Now what happens to the requirement to hold an ASIC?
The regulations pertaining to ASICs are nothing to do with CASA. They are merely one of many issuing bodies that can process applications.
How do the get an ASIC without an ARN?
I wouldn't see this as an insurmountable problem. All that needs to be demonstrated for the issue of an ASIC is an operational need. Other documentation or verfication by the school could satisfy this requirement. For example in the case of a non-pilot applicant, a signed declaration or letter from an employer is usually sufficient.