An american boat is implied by US customs to be one owned by a US citizen and if not customs registered has to be state or USVI registered.
It is not clear whether state registration requires citizenship but it does require proof of id. I suspect that if it is foreign owned ie by you state registration would not make any difference to your handling by customs, and these days you could find major problems arise.
In contrast to an american boat "The master of a foreign-flag or undocumented foreign pleasure boat must report its arrival to US Customs immediately and must make formal vessel entry on a Customs Forms (CF) 1300 within 48 hours. In the absence of a cruising license, vessels in this category must obtain a permit before proceeding to each subsequent US port."
While it talks solely in terms of the boat arriving, one would hope that a cruising permit could be obtained for a boat that is purchased there without having to leave first. Otherwise they have to obtain clearance before leaving for another country or US port.
There could also be complications with your visa. Unless you have certain ones, a 2b I think and another, giving multiple re-entry you cannot arrive by sea although you are from a country that is exempted a visa for short stays. Also there is a restriction on obtaining further entry by arriving from adjacent countries Canada Mexico and the Caribbean.
Normally a boat would be registered in the country of which you are a citizen, citizenship being a requirement of registry. The alternative of a flag of convenience means registering a company presumably as owner higher costs and annual fees.
There is no point in paying sales tax in the US when you will be hit by duty and GST on return to Australia. I think you may find it simpler to front up and register as an australian yacht. There are complications but probably less severe than if one wanders into the uncertain bureaucratic nightmare of not being the routine or usual case. You could find yourself deported.