I don't think it is feasible to move Australia Day, although it is regretably also Invasion Day for the Indigenous population. Another day of Australian National celebration would be good, sadly Federation took place on the 1st January...
There should probably be a national holiday for the anniversary of Reconciliation (we have something like the 2nd smallest number of holidays in the Western world), which is both a day of celebration, but also of sombre reflection - much in the same way as ANZAC Day.
I believe that the more pressing issure, compared to moving Australia Day is moving forward to becoming a Republic, and divorcing our current Flag. Also, moving forward to becoming a Republic will open up another date to celebrate Australia Day, a date which will neither seem arbitrary and sensless (which a random date would at the moment), and actually celebrates a shared milestone between our Indigenous and our (because we all are in one way or another) Immigrant population.