It was my impression that the Federal government tried and failed to pass legislation to let private Christian schools discriminate against LGBTQ students and staff. Your euphemistic language rises to the level of a lie when you try to say that it's about distinguishing them from private schools.
Thankfully, due to your recent federal elections, the ant-LGBTQ bigots who run some of those private Christian schools have little to no hope that laws will pass to enable their bigotry.
The Australian people have firmly shut them down.