Lots of helpful stuff from Ash above, good beavering!
It is possible that the child fell into wardship or similar at some point and progress to Australia went from there. The Birth Cert should be public record and relatively easy to attain if birth was in ROI. I do not know if records relating to court processes are public. There may also be records you are entitled to see in Australia. I'd definitely want to know everything possible about the situation as it stands before I made my first move. If the child was ward of the court and went to current guardians to avoid care proceedings (GB Terms). The basis of where we are now might be different.
It is possible that the child fell into wardship or similar at some point and progress to Australia went from there. The Birth Cert should be public record and relatively easy to attain if birth was in ROI. I do not know if records relating to court processes are public. There may also be records you are entitled to see in Australia. I'd definitely want to know everything possible about the situation as it stands before I made my first move. If the child was ward of the court and went to current guardians to avoid care proceedings (GB Terms). The basis of where we are now might be different.