Tony Abbott has accused the Labor party of using a “scare campaign” over doctor fees in its efforts to retain to Kevin Rudd’s former Brisbane seat of Griffith.

The prime minister argued he had no plans to introduce a GP co-payment, as he rallied hundreds of Liberal National Party (LNP) supporters at a campaign launch on Saturday, one week before the byelection triggered by Rudd’s resignation from parliament.

Abbott said he, as health minister in the Howard government, had worked closely with the LNP candidate Bill Glasson, then president of the Australian Medical Association, to improve the health system and deal with the medical indemnity crisis.

“I tell you what: you go a long way to find a better person than Bill Glasson,” Abbott told the gathering at the Easts Leagues Club in Coorparoo, an inner-south Brisbane suburb.

He said Glasson had the capacity to reach out to people of any political persuasion and provide strong representation, and was not going into parliament to promote his career or build his ego.

“I have never met a man with better or purer motives for going into the parliament of this country,” Abbott said.

But in the nearby suburb of Greenslopes, the Labor candidate, Terri Butler, was preparing to cut a cake to celebrate Medicare’s 30th birthday as she intensified the focus on a potential $6 co-payment for bulk-billed visits to a GP.

The idea – proposed in a submission to the government’s commission of audit – was advocated by Abbott’s former health adviser Terry Barnes, who argued the rising health costs were unsustainable to the taxpayer. Labor seized on Glasson’s initial support for an “affordable price signal” but the LNP campaign quickly sought to back away from the idea.

The health minister, Peter Dutton, and the treasurer, Joe Hockey, have refused to rule out the move. Abbott was asked by reporters on Saturday to guarantee he would not introduce a GP co-payment.

“Nothing is being considered; nothing has been proposed; nothing is planned,” Abbott said.

He lauded Glasson as “the best friend that Medicare has ever had” and denied he should categorically rule out the GP fee in order to scotch the Labor warnings.

“I think I have knocked the scare campaign on the head. This is all the Labor party has got. Nationally or locally all they’ve got is a scare campaign and I think that the people of Griffith and the people of Australia are bigger than that,” Abbott said.

Butler brushed off the scare campaign claims, saying she was talking to voters about important issues and fighting against moves that would “undermine our universal healthcare system and Medicare”.

She said the prime minister should have taken the opportunity to rule out a GP co-payment rather than using language that left the door open to it.

“We all know that those are weasel words,” she said after a barbecue with Labor supporters. “If they were going to categorically rule it out they should have done so today but instead there’s this no-plans language.”

Butler, an employment lawyer, said the LNP’s union-themed “personal attacks” were an attempt to distract from healthcare and education.

The foreign affairs minister, Julie Bishop, had earlier used a speech at the LNP launch to describe Butler as “yet another union puppet” who would represent “the interests of union bosses”.

Bishop said Glasson had a history of working with Abbott to solve problems by putting the community's interests first.

Other high-profile supporters attending the launch included the Queensland premier, Campbell Newman, the federal attorney general George Brandis and the outgoing veteran senator Ron Boswell.

Glasson, who is running for Griffith for a second time after reducing Rudd’s margin in September, told the gathering he offered knowledge, experience and an ability to develop practical solutions. He said doctors were good listeners and health was his passion.

Glasson attracted a swing to the LNP of 5.45% in September and requires a further 3% to win the seat from Labor on 8 February. The byelection is the Abbott government’s first electoral test since it swept to power. Both major parties argue that the result will be close but the betting markets favour Butler.