In an Australian context, should "Aborigine" be capitalised?

Yes. In the Australian context “Aborigine” and “Aboriginal Australian” should appear capitalized because they are acting as proper nouns referring to specific group. When being used to refer to indigenous people in general they would not be capitalized. For example if you were to write:

This is a fine piece of aborigine art.

you would be talking about a piece of art created by the original inhabitants of their respective region. But if you were you to write:

This is a fine piece of Aborigine art.

You would be talking about an artwork from the Aboriginal Australian culture.

Thus one could write:

The aboriginal people of Australia are the Aborigine.

And, if—for some reason—you were to write:

This is a fine piece of Aborigines art.

You would be referring to the Roman myth.

Oh my goodness, the person above has no knowledge of how to use the term correctly! You could get into big trouble for being so racially insensitive. An Australian here:

Aboriginal is the singular. Aborigines is the collective nown.

So the correct term is 'Aboriginal Art' not 'Aborigine Art'. Though it is grammatically correct, you could say this is 'an Aborigine's piece of art' but you would never refer to a person in such an objectified manner in practice.

The most important thing to be aware of is that the indigenous peoples of Australia are made up of HUNDREDS of tribes, each with their own diverse customs, culture and language. Therefore it is more appropriate often to refer to the tribe itself, rather than the generic collective label.

Eg. Wurundjeri elder, William Barak...