Labour peers launched an attack on the party leadership last night, accusing it of “paralysis” and “cowardice” over Brexit policy, as bitter recriminations over disappointing local election results burst into the open.

The row exploded as calls grew for a full post-mortem into why Labour failed to make more decisive progress in key council battlegrounds and lost control in others, despite the Conservatives being mired in crises over immigration, Brexit and the NHS. Although the party performed better in London than at any time since 1971, it was unable to take Tory strongholds in the capital, including Wandsworth, and fell short in several councils where it hoped to make gains, failing to win in Nuneaton, Derby and Basildon.

With Jeremy Corbyn increasingly under pressure from Remainers to develop a more distinct and robust pro-EU policy, Labour peers said they were outraged that the party appeared to be trying to block their efforts in the Lords to achieve a soft Brexit that would allow the UK to remain involved in the EU’s internal market, and provide a solution to the Irish border problem.

Q&A What is a soft Brexit? Show A soft Brexit, while not officially defined, would keep Britain in either the single market or the customs union or both. It could be achieved along the lines of the Norway model (see EEA/EFTA) or via an FTA, but would require concessions on free movement, ECJ jurisdiction and budget payments. Brexiters do not consider a soft Brexit as really leaving the EU. See our full Brexit phrasebook.

The Observer understands that more than 40 Labour peers are ready to back a cross-party amendment to Brexit legislation that would instruct the government to begin negotiating future UK membership of the European Economic Area (EEA).

This would mean that the UK would leave the EU and not be part of the common agricultural policy, common fisheries policy or be subject to the European Court of Justice. It would also be able to strike its own deal on freedom of movement, while being inside the EU internal market.

But the peers have reacted furiously after being told that the leadership is instructing peers who take the Labour whip to abstain in the vote on Tuesday, meaning it will not be passed. The peers say the majority of party members back a soft Brexit including single market membership and the measure would stand a good chance of being passed in the Commons if it were voted through the Lords.

The businessman and Labour peer Lord Waheed Alli, who has put his name to the amendment, accused the leadership last night of being “paralysed by indecision”. “This is complete cowardice. There is no point in being in politics to abstain, If you stand in the middle of the road someone is going to knock you over.” he said.

His fellow Labour peer Lord Cashman said it was essential to look at a solution that would allow membership of the EEA, as it would not only be good for UK business but also solve the Irish border problem which is now threatening to torpedo a Brexit deal. “Why the frontbench will not accept that as a negotiating position I completely fail to understand.”

The party and Momentum said the local elections represented progress. Laura Parker, Momentum’s national coordinator, said: “We’re very pleased to see healthy gains in areas where Momentum campaigned hardest. This just shows how important it is to have an engaged, enthusiastic movement – people who will get out into the community and spread Labour’s message.

“The untold story of the night is Labour’s advance on last year’s snap election result. If this result were played out at a general election, the Tories would lose 12 parliamentary seats and be unable to form a government.”

But Labour MPs said there were urgent questions the party needed to address that explained why Labour had not performed in line with expectations. Angela Smith, MP for Penistone and Stocksbridge and a member of the parliamentary committee that relays backbench views to the leader, said the row over antisemitism had to be addressed. So too did concerns in many local parties about centralisation, and resentment over attempts to purge moderates from positions in the party.

“We are increasingly being seen as a party that is ugly and unpleasant. Voters do not like the total control mentality. They don’t like bullying and they don’t appreciate attempts to shut out alternative views and debate.”

Yesterday party moderates launched a new campaign to hold a vote over the party’s Brexit position at its autumn conference, in an attempt to shift it towards a soft Brexit.

Tony Blair’s former director of communications, Alastair Campbell, said there was a “dreadful complacency about what it takes to win” at the top of the party and called for leadership on Brexit. “Frankly, if we cannot beat this shambles of a Tory party, we don’t deserve to be in the game,” he said.

“All my doubts relate to the course now being followed and, because it is such a huge issue that will determine so much of our kids’ futures, the approach being taken to Brexit … We in my generation have to do what we can to stop the madness of Brexit.”

Alastair Campbell called for efforts to stop ‘the madness of Brexit’. Photograph: Karen Robinson/The Observer

Ben Bradshaw, the former Labour cabinet minister, said: “I think we can stop Brexit and we must stop Brexit, but if we don’t do that we must make sure we stay inside the single market and customs union. Theresa May lost her majority for a hard Brexit in the election last year.”

After the Tories averted a meltdown of their vote as many former Ukip voters turned to them, pro-Brexit Conservatives were calling yesterday for Theresa May to honour the will of the people and get out of the customs unions or any version of it. Pro-Brexit ministers are already warning that May could end up causing an electoral “disaster” for the party if she keeps insisting that she wants a “customs partnership” with the EU.

Some cabinet ministers are said to be among those who fear that the plan, which was rejected by the Brexit cabinet sub-committee last week, would stop Britain making a clean break from the EU by the time of the next election, due in 2022. May is being warned that anything other than a proper exit from the customs union will see the party punished by pro-Brexit voters.

Writing on theguardian.com, Barry Rawlings, leader of Barnet Labour group, said the party had failed to win the borough because “so many of our neighbours believe we tolerate antisemitism”. He called on the party to reveal what action it was taking in relation to complaints of antisemitic behaviour, to demonstrate it is taking the issue seriously.

“It is agonisingly difficult to describe the feeling of standing as a Labour candidate in a political environment that has turned toxic,” he writes. “Some members didn’t want to put up posters for fear that neighbours might think they didn’t care about antisemitism. A lifelong member broke down in tears on the doorstep because this would be the first election when she didn’t vote. Doors were slammed in our faces and leaflets were taken out of our hands and torn up in front of us.

“We have lost trust with our Jewish brothers and sisters, and it will be a long road back to regaining it. But it’s a road we must travel if we want to claim to be a political movement that can represent any and every community in Britain.”