Cairns/Port Douglas or Brisbane/Sunshine Coast/Gold Coast for 4 days during the winter (June/July)

Thinking of doing a two week trip in the June/July timeframe. 3-4 days in Uluru, 3-4 days in Hamilton Island, 3-4 days in Sydney. I have about 4-5 days left to plan in my itinerary and I am thinking flying either to Cairns or Brisbane for the remainder. What do you guys recommend? Any activity recommendations or places to stay? Budget can go up to $600 AUD per night for three person family. Thanks!