In regards to your request, I have asked around and it seems like the New South Wales Department of Education would be your first point of contact as they would have contacts that are responsible for STEM in NSW. Please contact them through http://www.dec.nsw.gov.au/contact-us
Another organisation that may be of assistance is the Association of Independent Schools of NSW. They have a section on STEM with direct contacts: https://www.aisnsw.edu.au/Services/Partnerships_in_Education/STEM/Pages/default.aspx
As for Victoria (Melbourne), you can contact the Victorian Department of Education through here: http://www.education.vic.gov.au/about/contact/Pages/default.aspx
I hope you are able to find a STEM orientated school that meets your requirements.