You can marry legally in the US regardless of your immigration status. The K-1 visa is for those who intend to settle in the US. Since your fiance will not be adjusting immigration status, and will be leaving the country before the expiration of the allowed period of stay, you will have no problems.
Your fiance does not need a K-1 visa.
There's not much to support this on US government sites, other than the fact that discussions of K-1 visas invariably include discussions of how to go about adjusting status to permanent resident after the wedding. However, you can find immigration law sites that support this, such as http://www.visapro.com/green-card-family/getting-married-on-a-tourist-visa.asp
Articles such as this invariably discuss the risks for people who are intending to immigrate to the US after getting married, but since this is not your intention, these risks do not apply to you.
Furthermore, late in 2013, the US actually relaxed its policy to allow VWP nationals to enter the US on the VWP, marry a US citizen, and then apply for adjustment of status to that of a permanent resident. See http://immigrationview.foxrothschild.com/general-immigration-news-and-updates/visa-waiver-policy-change-makes-planning-life-in-the-us-much-easier/ for more information. Since your fiance would be allowed to enter on the VWP even if you intended to immigrate, he will certainly be allowed to enter given that your real intention is to return to Australia.
Given the relatively recent vintage of this policy change, it's likely that the information you've found on the internet that indicates otherwise is out of date.
As for the validity of the marriage, there should be no question, as even illegal immigrants can marry in the US. See, for example, http://www.alllaw.com/articles/nolo/us-immigration/can-undocumented-illegal-immigrant-marry-citizen.html. In articles like this, the validity of the marriage is never in question. The issue at hand is whether the non-citizen spouse can regularize his or her immigration status after the marriage.