Yes. There is no requirement to involve the Department of Human Services in your child support arrangements but until an assessment is in place, the Department cannot help enforce late payments.
You can make a private Child Support Agreement providing for child support to be paid above any assessed amount and include additional payments such as school fees, extracurricular expenses and medical costs.
A Binding Child Support Agreement should be drafted by lawyers as it has strict legal obligations and cannot be altered unless terminated by a further Agreement. It also requires that both parties to the Agreement receive independent legal advice prior to signing the Agreement. The Binding Child Support Agreement can be registered with the Department of Human Services who may help to collect payments of child support if required. They cannot enforce such things as school fee payments or other non-periodic expenses. Enforcement of non-periodic child support is by way of application the Federal Circuit Court.