Update: Looks like an ARB can be processed in 3 days, while a tourist visa can take up to 15, so this is not the way to go after all. Leaving this here for posterity though, since it's still the general answer to most "can I enter with the wrong passport" questions.
I can't find any definitive source on whether you're technically required to enter Brazil on a Brazilian passport if you have one, the Brazilian embassy to Australia merely says you are "expected" to use it. Nonetheless, the practical thing would be to apply for the visa for your Australian passport, and use that to travel to and from Brazil. Three things can happen:
- The visa application is refused because they realize you're Brazilian. Fine, you're still in Australia, follow gmauch's advice and apply for the ARB temporary passport instead.
- Your visa is granted, you fly to Brazil, and at the border they realize you're a Brazilian citizen. But you're already there and a citizen, so they can't deny you entry — welcome to Brazil!
- Your visa is granted, you fly to Brazil, you enter as as Australian. Problem solved, and this is by far the most likely scenario.
Disclaimer: In scenario 2, if using a foreign passport to enter Brazil turns out to be illegal, there's a chance that getting busted will involve getting held at the border for a while, fines or other unpleasantness. You could do worse than ask the embassy, foreign ministry or other reliable source for advice, although it may be difficult to get a straight answer out of them.
If you're unwilling to risk this, there's option 2B: get the visa, fly to Brazil as an Australian (so the airline lets you on board), but at the border, show them your expired Brazilian passport. This is perfectly legal, and they'll let you in eventually, although you may be in for a long wait while they figure out what happened and how you got here. And you'll now need to get a Brazilian passport before you can leave... unless you exploit the Mercosur loophole to leave Brazil with Brazilian ID, enter a neighboring country as Australian, and fly out from there.