My daughter was in a similar situation a few years ago. I posted about it in a post called “Mental Health and Study Abroad,” which I can no longer find. She went to France, not Australia, so the stakes were lower.
The upshot was that we chose to let our daughter go only if she took ownership of everything, from getting a semester’s supply of her medication (she was there a year, but came home for Christmas to refill), and arranging a therapist over there. She had to deal with the insurance company herself and basically demonstrate that she was able to manage her condition without our help. Of course, we had passports and were willing and able to fly over there if necessary.
She had a great time, starting with a fellowship in England the summer before her France program, and then a summer job as a tour guide in a cathedral.
For people with mental health challenges, college is a great time to start learning to manage the condition themselves.