If a training center says you need to get a FAA private certificate to get your FAA ATP I would remove them from the list of training centers you are looking at. The regulation is very clear. FAR 61.153 says -
"(d) Meet at least one of the following requirements:
(3) Holds either a foreign airline transport pilot license with instrument privileges, or a foreign commercial pilot license with an instrument rating-"
Your Australian license counts. You will have to have the Australian CAA verify your license to the FAA. You will need a TSA background check. You have to complete the ATP CTP classes. And you must pass the 737 type rating program. That is all. Most training centers have seen many foreign pilots and know the procedure.