The family's current bridging visas, issued in June, are set to expire on Wednesday.
Without the visas, Nades, Priya and their two Australian-born daughters could be sent back to immigration detention or community detention, or removed from Australia.
It comes after the High Court last month refused to hear an appeal on behalf of the family's youngest member, four-year-old Tharnicaa.
The family has been living in Perth after being released from years of detention - the last stint on Christmas Island - after Tharnicaa was medically evacuated from the island earlier this year with a blood infection that left her gravely ill.
The Murugappans were living in Biloela before Priya and Nades' bridging visas expired in 2018, after which the family was taken into detention during a night raid.
Nades and Priya have said they face persecution if deported to Sri Lanka. They fled their homeland after the country's civil war and came separately by boat to Australia.
The government has repeatedly said the family is not eligible to permanently settle in Australia.
Simone Cameron, of the Home to Bilo advocacy group, welcomed the "surprise decision" from Mr Hawke to grant new three-month bridging visas.
"But today’s outcome means the family’s situation has largely unchanged. They are still essentially trapped in Perth, in community detention along with little Tharni,” she said in a statement.
Angela Fredericks, also of the Home to Bilo group, said Priya and Nades "are terrified of what might happen to them if their visas expire and they are ever forced from Australia to Sri Lanka".
“The majority of Australians want the Morrison Government to let this family return to Biloela. We know this family is safe in Biloela. No one can guarantee their safety if they are forced to Sri Lanka,” she said.
The meaning of 'close of business' was central to Thursday's court action, with the deadline for the family to make submissions to the minister being the subject of contention.
Barrister Angel Aleksov said lawyers for the family were invited in June "to give submissions on the minister's proposed course ... in respect of bridging visa applications".
The department is arguing the submissions were required by close of business, but solicitor Carina Ford's email with the submission was sent at 5:45pm Australian Eastern time, which would have been 3:45pm in Western Australia, where the family was based.
"The communication to them (the family) left it fairly unclear as to precisely when the deadline was, and also perhaps more importantly, where to send the correspondence with the submissions," Mr Aleksov told Judge Heather Riley.
"There was correspondence earlier in the week that my clients were given and they indicated a response would be made."
The court heard the minister was briefed at 6.58pm the same day and a decision made at 7.20pm.
Mr Aleksov said there was a "measure of unreasonableness in th e way that the department rushed into a decision without checking to see whether there was in fact a submission put to them". This was concerning given the long history of engagement between the parties and their representatives.
"We submit that was both unfair and unreasonable," he added.
The hearing will continue on 4 October.