Sumaya Bhuiyan, a 16-year-old teenager with autistic spectrum disorder, could be deported from Australia as soon as Friday.
The family's application for permanent residency was rejected when Sumaya failed a mandatory health check - an element of the process designed to avoid Australian taxpayers footing the bill for expensive medical procedures.
Her parents, both practicing GPs, have said they can financially support their daughter without the support of the state, with the exception of her attendance at a special needs school.
They appealed to immigration minister Peter Dutton for a "ministerial intervention" to override a tribunal decision.
But the family has now received a reply from the department saying Mr Dutton's assistant minister Alex Hawke had "personally considered your case and has decided that it would not be in the public interest to intervene".
Sumaya's mother told SBS she was surprised at the decision, given her status as a skilled migrant currently employed as a doctor.
"I couldn't accept this decision," said Dr Nasrin Haque. "When I found this email [from the family lawyer], I was shocked."
Sumaya failing the health check means the whole family is now inelligible for permanent residency.
Dr Haque was able to extend her own skilled working visa. Her 14-year-old son Sakir, Sumaya's brother, will be allowed to stay with her as a dependant.
But Sumaya's disability rules her out of this visa. The family has been told her visa expires on January 27, and she will then need to return to Hungary, where her father lives, and the family has citizenship.
Sumaya's parents are not sure how well their daughter understands what the decision means, but say the anxiety is having an impact.
"If something changes, her behaviour also changes. She gets more agitated, more anxious," said Dr Haque.
Her father Dr Shafiqul Bhuiyan, who works as a doctor in Hungary, is in Australia visiting his family.
Now, it seems all but certain he will be taking Sumaya back to Hungary with him on the return flight.
"I don't know what to do with her alone, really," said Dr Bhuiyan.
Sumaya is supported by her mother, brother and grandparents in Sydney, and Dr Bhuiyan is concerned he won't be able to give her the same level of care on his own in Hungary.
Sumaya does not speak Hungarian, and Dr Bhuiyan said all the country's special needs schools are taught in the national language.
Assistant Minister Alex Hawke was not available to comment on the specific case.
But his department provided a statement: "The Minister cannot be compelled to exercise his powers and he is not required to explain his decisions on any case," it read.
"What is or is not in the public interest is entirely a matter for the Minister considering each case on its own merits."
The department told SBS the minister only intervenes in a small number of cases, when there are exceptional circumstances.
Around six months ago, Mr Hawke did intervene involving an autistic child of highly educated Bangladeshi migrants, allowing them to remain in Australia.
The family is still hoping for a last-minute reversal of Mr Hawke's decision.
"I hope, I trust, he can reconsider our situation," Dr Bhuiyan said.
