Labour has said the Commons should be able to vote on whether to hold a second referendum in an amendment the party submitted on Monday night to Theresa May’s Brexit update.

It is the first time the party has asked MPs to formally consider a second poll, although the carefully worded compromise amendment did not commit the party’s leadership to backing a referendum if such a vote were to take place.

The wording called for May’s government to hold a vote on two options – its alternative Brexit plan and whether to legislate “to hold a public vote on a deal or a proposition” that is supported by a majority in the Commons.

The intervention came as the party’s leadership seeks to deal with divisions between Jeremy Corbyn and some of the leader’s closest allies who are sceptical about a second referendum and those who are more enthusiastic such as Brexit spokesman Sir Keir Starmer.

The party’s alternative Brexit plan, which would be the subject of a separate vote if the amendment were carried, proposes that the UK remain in a post-Brexit customs union with the European Union and have a strong relationship with the single market. Citizens’ rights and consumer standards would be harmonised with the EU’s.

Corbyn said: “Our amendment will allow MPs to vote on options to end this Brexit deadlock and prevent the chaos of a no-deal. It is time for Labour’s alternative plan to take centre stage, while keeping all options on the table, including the option of a public vote.”

But second referendum campaigners were last night gearing up to criticise the alternative Brexit plan. The People’s Vote campaign said that it was one of a series of Brexit promises – like no-deal – that were being made with “little regard for what could actually be delivered”.

The cross-party group, supported by dozens of Labour MPs, posed five questions that it believes Corbyn’s Brexit model struggles to answer, escalating tensions between campaigners and the party’s leadership.

A 16-page evaluation placed Labour’s plan on a par with both the no-deal option supported by hard Brexit Tories and the Norway-plus customs union alternative, in that they were no better than pledges made by the leave campaign at the time of the EU referendum.

“All three alternative plans,” the analysis added “share many similarities with the arguments made for Brexit in 2016 when big promises were made with little regard for what could actually be delivered.”

It quoted an interview with Corbyn in which he said he would be opposed to any deal that would make the UK poorer, and then cited a report by the National Institute of Economic and Social Research that said “every form of Brexit will damage the UK economy”.

The analysis also asked whether the UK would “have any say over its trade policy” under Labour’s policy of remaining in a customs union with some decision-making power over future EU trade agreements.

The document, whose authors are not named, said “it is almost inconceivable that the EU would agree” – citing the EU’s Lisbon treaty – and that “being in a customs union without a say would be very damaging” because the UK would be force to liberalise markets to third countries on a non-reciprocal basis.

Tensions within Labour have been rising as MPs who support a second referendum want to pressurise Corbyn into supporting another Brexit vote, in line with the party’s policy to explore it as an option if it cannot secure an early election.

A total of 71 Labour MPs declared they supported a second referendum last week on the morning of the day the party lost a vote of no confidence in Theresa May’s government, meaning that an early election was unlikely.

The group is planning to take the argument further at a news conference on Tuesday, fronted by Labour MPs David Lammy and Bridget Phillipson with Liberal Democrat deputy leader Jo Swinson and Caroline Lucas, the Green party only MP.

Labour backbencher and second referendum supporter David Lammy said the amendment represented “a big step forward” but added that the party’s alternative Brexit proposal “deserves to be properly scrutinised” now that Corbyn wanted it to be centre stage.

• This article was amended on 22 January 2019. An earlier version incorrectly referred to Caroline Lucas as the Green party’s leader in Westminster. She no longer has that role, however she is currently the party’s only MP.