Bankstown - Parkes - Broken Hill - Coober Pedy - Ayers Rock, would be my choice.
Easier legs in the 182, card swipe fuel, not so much remote area, better coms and more choices if you have a problem. Plus lots of things to see enroute (Broken Hill, Flinders Ranges, Coober Pedy, Lake Eyre all must see destinations)!
By all means return via Birdsville, but I would suggest Bourke would be a better fuelling stop (last times I got fuel at Tibooburra it was a real pain in the arse).
Remember Ayers Rock to Birdsville takes you over the Simpson Desert. It is a fair haul in a 182 and not many options to land in between. So take precautions like carry some water just in case.
Forget about tankering gas! On this trip you don't have that many options, pack a smaller o'nite bag and take the maximum fuel your weight will allow (should not be too much problem in a 182)!
And the advice regarding the weather is good it can be pretty hot and bumpy that time of the year. Go early and go high. (And if your pax are not frequent flyers ensuring you have some discreet sick bags could be a good idea)
Enjoy, its a great trip... although December would not be my choice.