Theresa May’s “you called me nebulous” outburst at Jean-Claude Juncker in Brussels on Friday came at the end of a week that was extraordinarily challenging even for a politician with her legendary resilience.

In the space of five days she was cajoled by her cabinet into pulling the meaningful vote, forced to cut short a whistle-stop tour of EU capitals to plead with her own colleagues to let her stay in Downing Street, and then suffered the indignity of being told by EU leaders that they did not know what she wanted.

Her steadfastness in the face of extraordinary personal attacks and an unremitting schedule of hard graft is undeniable. But some of her colleagues believe it is also her greatest weakness.

At moments when a different kind of politician might flex with the mood in her party or in parliament, May tends to plough relentlessly forward, convinced others will ultimately come round.

For a few brief hours on Wednesday it looked as though her own MPs might have had enough, but after weathering the no-confidence vote she now has the chance to submit her strategy to its ultimate test and try to get her deal through parliament.

There is still – just about – a narrow, rocky path to success. While the EU27 rebuffed her demands for reassurance on the backstop in Brussels, work will go on to find a form of words that could win over at least some of her party’s recalcitrant Brexiters, convincing them the backstop is not a dastardly trap.

The minds of some may be focused by the sense that the option of a second referendum is gathering support, and that a majority in parliament could probably be cobbled together for a softer form of Brexit – though not for “super Canada”, as Boris Johnson calls it.

Perhaps a few days at home, hearing the views of friends and constituents exasperated by politicians’ failure to implement the result of the 2016 referendum, will persuade others that May’s deal would be better than no Brexit at all.

May has tried to make the argument to her party that they must choose her Brexit or no Brexit. That looks more true now that the Brexiters have tried and failed to remove her – unless they are sufficiently in wrecking mood to want to bring down the government altogether.

Anand Menon, director of UK in a Changing Europe, said: “I think things can move, and I think they can move in her favour in her own party.” He doubts it will be enough to secure a majority for May’s deal at first pass. But a narrower margin of defeat could help to persuade at least a handful of Labour MPs whose voters do not want them to “block Brexit” to support her at a second time of asking.

If May fails to win parliament’s approval for her deal even at a second pass, only radical options are available. She cannot now face another vote of no confidence from Tory colleagues – the ERG has played that card already.

She could step aside, or throw open the doors of No 10 and invite Jeremy Corbyn to cooperate in drawing up an alternative Brexit blueprint that would satisfy enough MPs to pass.

But neither of those would sit easily with her rigid political style and her strongly held belief that splitting her party would be a disaster.

Stephen Fisher, an elections expert at Oxford University, argues that May’s least bad option at that point may be to reach for a referendum, pitting Brexit on her terms against remaining in the EU.

“Theresa May should want to persuade the people to back her deal if parliament won’t because she believes it is the best thing for Britain,” he said.

The prime minister firmly believes her deal, with its promise of controlling immigration, is closer to the public’s understanding of what they voted for in 2016 than Norway-plus, for example, which is why she has repeatedly sought to make the case for it directly to voters in recent weeks.

The Tories would be bitterly divided, but she may hope forcing Labour to become the party of “Remoania” would mean the poll did more damage to Corbyn’s party than her own.

Indeed, by pledging a second referendum herself, upfront, before the meaningful vote, May could even secure the backing of remainers, who would hope to win it, and at the same time sharpen the choice for Brexiters between her deal and not leaving at all.

Some of her cabinet colleagues now believe this could be the only way to smooth the path for her deal, though the parliamentary mechanics of it are challenging to say the least.

A snap general election, by contrast, could leave the way open for Labour to run on a constructively ambiguous ticket of negotiating its own version of Brexit and perhaps then putting it to the public in a referendum, Harold Wilson-style.

Avoiding a split in her own party has been a guiding principle for May, but she believes – “with every fibre of my being”, as she has put it – that her deal is the right one.

She doesn’t want another referendum; but if all other options have been exhausted, as Fisher put it, “it may well be that she should do what she thinks is right and hope her party can heal itself after she has gone”.