A row has broken out among campaigners for a second referendum about when to push the issue to a vote in parliament, with the Conservative MP Sarah Wollaston resisting pressure not to table her amendment demanding a “people’s vote”.

With the Labour leadership withholding its support, some campaigners fear forcing the issue to a vote on 11 December would undermine their cause.

They believe once it has been shown that there is no majority for a second referendum – and achieving one is likely to be impossible without Labour backing – it will be difficult to return to the question again if May’s deal is rejected.

Wollaston said she remained a “passionate supporter” of a people’s vote, but would wait until after the weekend before deciding whether she would table her “doctors’ amendment”.

“We are listening to concerns that are being put to us, that it would fail badly, and that might be used as an excuse,” she said. Some MPs fear the government could then say parliament had definitively rejected a second referendum.

“The reality is that it can’t succeed unless the Labour frontbench support it. Genuinely there are two schools of support here about whether it’s a good thing to keep the pressure on,” she said, adding: “It’s a strategy question.”

However, one senior member of the People’s Vote campaign claimed Wollaston was being “sat on”, and there had been a bitter row about the best way to proceed. “It’s like world war three,” he said.

Meanwhile, the Liberal Democrats are drafting their own amendment on a people’s vote, as an “insurance”, in case Wollaston decides to hold back.

Facebook Twitter Pinterest Sarah Wollaston. Photograph: Michael Bowles/Rex/Shutterstock

The government’s announcement that it will accept amendments to the motion approving May’s Brexit deal – with up to six to be voted on, before the deal itself – has sparked a scramble to decide which questions to press.

The Brexit select committee chair, Hilary Benn, laid down his own cross-party amendment on Thursday, signed by other chairs including Rachel Reeves and Dominic Grieve. It aims to show that the House of Commons rejects the idea of a no-deal Brexit – and allow MPs to take control of the next steps if the deal falls.

Labour is likely to back the amendment, with the shadow Brexit secretary, Keir Starmer, repeatedly insisting he wants to take a no-deal outcome off the table.

Meanwhile, the Conservative former minister Jo Johnson warned on Thursday that May’s Brexit deal could lead to electoral Armageddon for his party.

In his first major speech since resigning as a transport minister earlier this month, Johnson joined David Willetts and Justine Greening to sketch out a future in which the party faced an existential crisis and would have its brand thrashed by the economic fallout from Brexit.

Johnson described the package their party leader had agreed with the EU as a “botched deal” that would put British firms at a competitive disadvantage and fail the services sector, which he said had been “scandalously” neglected during negotiations on Brexit.

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He added: “Brexit is seen as a project driven by the Conservative party, and this half-baked, worst of all worlds Brexit could trigger an electoral defeat on the scale of 1997, or worse, with this ‘Tory Brexit’ label an albatross around our necks for years to come.”

Such an outcome would “roll out the red carpet for Jeremy Corbyn” and lead to “communist ideologues” coming to power in Britain.

Sketching out a plan for another referendum that would take place by the end of May 2019, Greening said article 50 could be extended by another four months.

The former education secretary said she was backing a second referendum because the country was tired of what she described as “backroom deals”. She said it was incredible to see May “touring the country and talking to people who she did not want to give a vote to”.