Baby with American citizen father, travelling to America

I am an American citizen, living in Australia, with my wife (Australian citizen) and baby girl (born here in Australia).

We are travelling to America in a few weeks. We have obtained an Australian passport for my baby girl, hoping to use it for her when entering the US.

However, my understanding is that

  1. Any child born with an American parent is automatically an American citizen (at least until they reach 18 years of age) and
  2. America does not recognise dual citizenship, requiring you to use your American passport to enter the country, if you have one.

Does this combination mean that I am required to register my daughter with the American consulate here in Australia, and obtain an American passport for her, before I am able to travel to America with her?