I spent two weeks in Sydney over Christmas. It was a lot of fun!
The best way to go to Japan as a diplomat would be to talk to your Foreign Service:
Australian Department of Foreign Affairs and Trade homepage
Your aunt could have indeed, if she had a rinji menkyo (臨時免許) which is a temporary teaching license. If you are serious about teaching in Japan full time for an extended period of time, you will need a kyouin menkyo (教員免許) which is what I am earning. For that you need your Aussie certification first.
I could walk you through the process, but you'd have to make a decision about your career path, because being a diplomat and being a teacher are two very different paths.