If the trip conflicts with his visitation then you will need his permission. If that's the case, I strongly suggest working with him to adjust the schedule and make sure he gets his parenting time in another way.
She is incorrect, a health care POA does not allow her to make decisions or be the "authority" that everyone has to go through to contact your father. She is confusing it with Guardianship/Conservatorship (although from the sounds of it I wouldn't point her in those directions). A health care directive is a narrow form which only allow the agent to make health care decisions for the other person (your father).
Overall, this sounds like a very ugly situation that's only going to get worse....
So long as she is a minor she will be subject to the parental laws of Australia. I don't know Australia's laws, but I imagine your concerns are correct. They very likely have kidnapping/custodial interference laws to protect children there, and he would certainly be violating them by taking a minor out of her country to live with him.
Her solution likely involves getting the Australia equivalent to child protective services involved to help her and if necessary the juvenile courts of Australia.