FAA validation start page:
https://www.faa.gov/licenses_certifi..._verification/
61.75 states:
§61.75 Private pilot certificate issued on the basis of a foreign pilot license.
(a) General. A person who holds a foreign pilot license at the private pilot level or higher that was issued by a contracting State to the Convention on International Civil Aviation may apply for and be issued a U.S. private pilot certificate with the appropriate ratings if the foreign pilot license meets the requirements of this section.
(b) Certificate issued. A U.S. private pilot certificate issued under this section must specify the person's foreign license number and country of issuance. A person who holds a foreign pilot license issued by a contracting State to the Convention on International Civil Aviation may be issued a U.S. private pilot certificate based on the foreign pilot license without any further showing of proficiency, provided the applicant:...
So in short, you need an ICAO PPL or higher to be issued a "based on" FAA PPL without further tests.
The first link contains an e-mail address. I suggest you contact them to see what needs to be done in your case. I suspect you'll need to do the FAA PPL written and FAA PPL practical exam (plus maybe some prep, plus an FAA class III medical) to be issued a standalone FAA PPL. Which, for the long term, is a better proposition than a "based on" FAA PPL anyway.
Also, read the link about Australia at the bottom. The problem is with privacy laws as they are, the US FAA cannot simply contact the AUS CAA and ask them if your license is valid. You need to give the AUS CAA permission to divulge your details to the US FAA.