Jeremy Corbyn’s letter to Theresa May last night offering to slightly soften his Brexit stance risks stirring a backlash from Labour supporters. A new poll makes it devastatingly clear that if Labour is perceived as having enabled Brexit, it will lose 45 seats – including five of its seven precious Scottish seats.

Labour’s true heartlands are remain: four-fifths of Labour voters think May’s deal will hurt the British economy and 91% of them don’t trust the government to deliver a good Brexit. Remainers will desert and punish the party in droves unless it opposes Brexit. This outlook comes from a large poll of more than 5,000 people – it was commissioned by TSSA, the transport union, which has always backed Corbyn. It was meant to be private advice to him, but it leaked. “There can be no disguising the sense of disappointment and disillusionment with Labour if it fails to oppose Brexit, and there is every indication that it will be far more damaging to the party’s electoral fortunes than the Iraq war,” the polling document reports.

“Labour would especially lose the support of people below the age of 35, which could make this issue comparable to the impact the tuition fees and involvement in the coalition had on Lib Dem support.”

The letter Corbyn wrote to May seems to give ground, appearing to offer compromises where Labour might support her, dropping some of its reddest lines. In truth it doesn’t, because May couldn’t agree to most of it – especially the permanent customs union and close alignment to the single market, its agencies and regulators. Some Labour people claim he’s playing a clever game, craftily appealing to the party’s leave voters by appearing not to oppose Brexit, but actually making it impossible for Labour to support any Brexit deal May can put on the table.

That’s too clever by half, the TSSA poll suggests. Appearances matter. Labour voters need to hear their party’s full-throated opposition to Brexit and all its fiendish works. The time for prevaricating, fence-sitting and playing both ends against the middle is drawing to an end. Bad mistakes were made when Corbyn dashed to demand article 50 be triggered the day after the referendum, with Labour having no more clue than the Tories what shape of deal was possible. Besides, his demand that UK law stays aligned to all future EU worker and environmental rights looks plain gullible: the next Tory leader will be a Brexiteer who would repeal it.

It’s never too late to do an about-turn and there has never been a better excuse: look at the Brexiteer chaos, look how abominably badly May has negotiated, look at the risk to Ireland and the Good Friday agreement, and fear the reality of no-deal horrors. Labour can say it tried to find a way – but every available Brexit will make Britain weaker, poorer and meaner. Find the words – at last – to embrace all the positives of belonging to the world’s biggest trading bloc, among friends who stand together in a darkening world as the best available beacons of human rights and democracy. Is it in need of reform? Of course – what human institution isn’t? We should return to the EU, stamp out Brexitism and the whole rightwing agenda that sails in it.

That might be a stretch for Corbyn and his small coterie of Unite advisers, at odds on Europe with the trade union movement and the rest of Labour. But what he deeply thinks and feels hardly matters at this point. What matters is what he says and does now, if he wants Labour to win.

Some have hailed his letter as a cunning manoeuvre: just as May arrives in Brussels today, he has pre-empted her with a Norway-style alternative that could, just possibly, squeak through the Commons. EU negotiators could embarrass her by urging her to take it up.

He could find backing among those MPs of the Lisa Nandy/Caroline Flint/John Mann wing of the party already threatening to back a Brexit deal. The TSSA poll finds that Labour would lose 11 leaver seats by opposing Brexit, instead of 45 if it supports it. Those MPs in leave-voting seats claim they can bring 60 Labour MPs to back a softened Brexit. Others say no more than 25 MPs – the number voting with the government or abstaining last time – would cross the floor on the next vote. If Labour votes were to swing Brexit, history will look back on these as appeasers and trimmers who helped cause a national disaster – and a disaster for Labour too.

Timing is all: the moment for Labour to call for delay and a final-say referendum will come if May fails on a vote a second time. Wait until all that’s left is no-deal or a referendum. Meanwhile, Corbyn’s letter sends out a badly misjudged message and unless this tone is corrected sharply, expect a backlash from Labour members and voters.

Brexit is a Tory beast that Labour needs to skewer. The TSSA poll shows that any flirtation with Brexit will cripple Labour and alienate the young who Corbyn drew in. That’s confirmed by a new ComRes poll for Lead Not Leave, finding 72% of voters aged between 18 and 34 back staying in a reformed EU, compared with just 38% of people aged 55 and over. The young are even more strongly against a no-deal Brexit.

Meanwhile, the TSSA polling concludes: “If there is an election in 2019, Labour will get a lower share of the vote in every seat in the country if it has a pro-Brexit policy than if it has an anti-Brexit position.”

• Polly Toynbee is a Guardian columnist