Just bear in mind the FAA will only issue a PPL based off the foreign 'base" licence no matter what the class of licence held on your base licence CPL/ATPL etc.
The process is straight forward, take your Aus licence/medical/logbook into a FSDO in the US and they will issue you a temp certificate on the spot.
It used to be fairly straight forward with CASA and recognition of foreign ratings, just produce a record of your training and evidence of the rating on your foreign licence, pay a fee and CASA will issue the licence.
As for the bureaucratic dogs breakfast that we all know as CASA I'm sure this process is now an absolute nightmare!!!