The Denpasar District Court began hearing the appeal, known as a judicial review, on Wednesday, but after only 15 minutes the prosecutor asked for the appeal to be thrown out, saying the legal grounds it was based on were wrong.
However, the panel of three judges did not agree with the objection and Chief Judge Putu Suika said all parties would be summonsed back to the court on August 26.
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Outside the court, one of Rush's lawyers Frans Hendra Winata told AAP that when the judicial review resumed next week, the defence would ask for Rush's death sentence to be reduced to 15 years in jail.
However, they would also be satisfied if the judges reinstated Rush's original sentence of life in prison.
Rush, 24, did not appear in court on Wednesday, which his lawyers explained was because the head of Bali's Kerobokan Prison had not been authorised to bring him to court.
"What has been said by the head of prison is that the one authorised to bring the convicted is the prosecutor," Rush's lawyer Robert Khuana said.
Khuana asked the judges to order the head of prison to bring Rush to court for the August 26 hearing.
There was a small group of Rush's supporters in court on Wednesday, including his father, Lee, but they all left straight away after the adjournment was announced.
When prosecutor IB Chandra asked for the judicial review to be thrown out, he told the court he objected to the trial due to the way Rush was summonsed to appear in court using Article 152, which only applied to regular trials - not an appeal or a judicial review.
"We also object to the court appointment of judges because it was based on a pretrial case, not for judicial review cases," he told the court.
"This is a fatal mistake, not just some administration mistake, because it is about legal ground."
Chief Judge Suika called for a 30-minute recess so the judges could consider the prosecutor's objections, and when the hearing resumed the judges said they did not agree with the objections.
"To the prosecutor's objections, judges did not agree with it.
The trial will continue as has been decided by the chief of Denpasar District Court," one of the judges, Puji Harian, told the court.
