Yes you could Visit the USA with her having a tourist visa, but cannot stay in the USA and adjust status, a proper immigrant visa will be needed, either IR-1 or K-3. Consider the CR-1 it does not take much longer than K-3.
You register the birth at nearest US consulate, they will then issued a US passport for child.
http://travel.state.gov/law/info/overseas/overseas_703.htmlToo bad you overstayed in Australia, if you were there with residency you could have filed for the spousal immigrations visa directly to the US consulate, and saved a bunch of time as compared to filing for in in the USA.