I'm sorry if this isn't what you want to hear but I still feel that the two of you are in crisis as opposed to it being a one sided match.
Have you contacted the local hospital for an Outpatient's referral?
Have you also checked your status with the Dept of Immigration?
Have you also checked your children's citizenship status? (Children born to NZ parents are not automatically entitled to citizenship...which could work in your favour).
Citizenship by birth—from 1 September 1994 to 26 February 2001
From 1 September 1994 to 26 February 2001, a child born in Australia to a New Zealand citizen parent who held an SCV, is an Australian citizen by birth.
Citizenship by birth—from 27 February 2001
A child born in Australia on or after 27 February 2001 to New Zealand citizen parent/s is not an Australian citizen by birth, unless the New Zealand citizen parent:
Note: Children born to a New Zealand parent who was living in Australia temporarily as a diplomat or as the holder of a special purpose visa (that is guests of government, aircrew or armed forces and their families) are not Australian citizens by birth.
- was not affected by the 26 February 2001 changes
- held an Australian permanent resident visa or Australian citizenship.
You might wish to do this in relation to legislative requirements for NZ citizens claiming citizenship in order to be eligible for social security because people have obtained emergency citizenship in certain situations.
As for removing the children, alas, you must go through the legal system to be able to take them overseas. This is only fair to both parents.