Once I am complete I will apply to the FAA to convert it
You cannot "convert" your AUS PPL into an FAA PPL. You can only get an FAA PPL issued "on the basis of" an AUS PPL.
Technically that process can be fully done outside of the US but it's incredibly cumbersome and it needs an FAA designated-something person to help you close to where you live. The most practical way of doing things is to sort the paperwork from abroad, but nominate an FSDO in the US to handle the final phase. Obviously the FDSO that you nominate should be at the place where you'll be arriving/staying in the US. Your first port of call upon arrival is that FSDO then, who will issue you with the temporary "based on" PPL. The final one (the laminated plastic card) will arrive in the mail in a few weeks.
The other thing to consider is that a "based on" FAA PPL is useful for a flying holiday, but not much more than that. If you intend to fly N-regs regularly, it's much better to get a standalone FAA PPL. With a "based on" FAA PPL you're building a house of cards.
There's actually a bit more involved than just a quick circuit with some stalls and steep turns thrown in. For your FAA PPL ("based on" or "standalone") to be valid, you need to have done a BFR in the last two years. By law, the BFR is *at least* one hour groundschool, and *at least* one hour in the air. And if the instructor thinks you're not up to scratch, he can extend that time as required, until you're at the required level. (Technically it's not an exam, so you cannot "fail", but the instructor will not sign you off until you're at the "pass" level.)
AFAIK most people who show up in the US for their first BFR take quite a bit longer than those two hours. Not because of basic aircraft handling, but because of the different environment: legal, charts are different, flight services are different, R/T is different (and much faster and accented than what you might be used to) and so forth.
Any FAA instructor (a "CFI" in FAA terms) can do a BFR. So if you want to keep your "based on" or "standalone" FAA PPL valid while in AUS, you can simply find an FAA-licensed instructor in AUS and do the BFR with him/her.