I was in a similar situation but with the customs dept when I had to import personal effects into Thailand back in 2012. A fee was paid without receipt and another with receipt.
My advise if you can -front the cash -do it everything has its price.
Marriage certificate and birth certificate of your son are both required. Medical certificates to justify his condition. In all fairness I believe you will succeed easier once the covid situation eases up a bit. So you would probably need to stay a bit longer in Australia. All documents need to be attested by the Australian MFA and the Thai embassy in Australia and also the Thai MFA and Australian embassy in Thailand as per the procedures. Your own visa would be Non O as well as your son’s. There are financial requirements for bank deposits in Thai bank accounts which I am not sure about. My opinion is that while you prepare the documents you visit Bangkok when it’s opened up and engage a local law office to help you with all this. While hard it is definitely doable.