Since you're making a consular record of birth abroad, presumably your daughter was a US citizen at birth. She can stay in the US as long as she wants. While she will get a stamp admitting her for 90 days in her Australian passport, this doesn't apply to her, as she is a US citizen.

Technically speaking, as a US citizen, she must enter the US using a US passport. However, breaking this law has no real penalty. [See this answer, for an explanation of what could happen at the border](https://travel.stackexchange.com/a/78878/28009).

So no, she neither needs to fly back to Australia, nor return to customs after arriving.