Question regarding child support payments

My wife and I have recently separated ( legally can’t divorce till another 6 months of separation) . Long story short I’ve been in a financial abusive marriage from almost day dot , we have 3 children ( 10,9 and 5) . She has not worked in 10 years insisted to be a home stay mum . Anyway she has requested we do a private arrangement with regard to child support payments and wants it all to be amicable. My concern is if I start to do that and then either myself or her decides to go down the centerlink route what protection do I have that centerlink won’t chase me for money for the period that we had the private arrangement?? If I ensure all my bank transfers to her are clearly stated as child support and is that enough ?

Also I have used the child support calculator off the service Australia website and used that as a base of what my child support payments should be , if my calculations are wrong and I have been slightly under paying will that come back as missing payments even if we had an agreement on repayments ? If I have over paid I am not concerned about credit.