Lindsay Sandiford was sentenced to death in January after cocaine with a street value of $2.4 million was found in her suitcase as she arrived on the resort island, a shock verdict after prosecutors recommended 15 years in jail.
The 56-year-old lost a first appeal against the sentence at Bali High Court earlier this month and her lawyer, Fadillah Agus, on Tuesday lodged a last-ditch appeal to the country's top court.
"I lodged an official notification to appeal to the Supreme Court through the district court in (Bali's capital) Denpasar," he told AFP.
It came after three Court of Appeal judges in London on Monday upheld a previous court ruling that the British government was not obliged to pay for an "adequate lawyer" for Sandiford.
The court heard that she needs about £8,000 ($12,200) to continue her fight against her death sentence, and only £2,000 had so far been raised. The Foreign Office has refused to pay as a matter of policy.
But Agus pledged to continue defending Sandiford and said he hoped her family and friends could raise enough funds to cover the legal fees.
"This situation will not change my position to defend Lindsay, because I don't defend her only for the money," he said.
He also lashed out at the British government: "I can't comprehend it, because even Indonesia tries to defend its citizens when they are facing serious problems overseas".
The lawyer said a document would be submitted within 14 days outlining the grounds for her appeal, after which the Supreme Court would likely make a decision at a closed hearing in two to four months.
If she loses the appeal, she can seek a judicial review of the decision from the same court. After that, only the president can grant her a reprieve.
In a statement issued by legal charity Reprieve ahead of Monday's ruling, Sandiford, who claims she was forced to transport the drugs as her children's safety was at stake, said she was "desperate".
"If I should die -- and I hope I don't, but I fear I may -- then I hope that my execution will prompt the British government to do more for others," she said. Executions in Indonesia are carried out by firing squad.