Whoever is elected Tory leader and becomes prime minister will find the clock is still ticking. Britain is to exit the European Union on Halloween, 31 October. To avoid that outcome, the EU must agree to a further extension so that the government can alter the exit date. The new prime minister will also be confronted with the unforgiving arithmetic of the House of Commons. Most MPs oppose a no-deal Brexit, though Labour’s attempt to enshrine this in law failed yesterday. But MPs have also three times rejected the only seeming alternative: the withdrawal agreement so laboriously negotiated by Theresa May. The EU has insisted that no alternative agreement is available.

Two candidates for the Tory leadership, Dominic Raab and Esther McVey, propose breaking this deadlock by proroguing parliament – ending the session – so that MPs are unable to block a no-deal Brexit. This, some have objected, could involve the Queen – since prorogation, unlike the dissolution of parliament, which leads to a general election, is governed not by statute but by royal prerogative. Indeed, Lord Armstrong, the former cabinet secretary, has suggested that the Queen, before exercising her discretion on so contentious a matter, would consult with Jeremy Corbyn, as leader of the opposition, and perhaps also with others.

But the Queen would almost certainly be guided by the rule that has served her so well during her long reign. That rule requires her to act on the advice of her prime minister. Then, if there are objections, they will be directed at the prime minister, not the Queen. As such, a prorogation would be relatively straightforward.

The real objection to proroguing is quite different. It would be more than controversial. The word “unconstitutional” is perhaps too frequently used. But proroguing parliament, under the circumstances envisaged by Raab and McVey, would stretch the constitution to its outermost limits, if not beyond. A Brexit-led prorogation means delaying the new session of parliament until November. Normally this session would be expected to begin as soon as the new prime minister takes office, especially as Corbyn would almost certainly table a no-confidence vote directed against the minority government. In any case, a new session of parliament begins with a Queen’s speech, which would be subject to an opposition amendment ruling out a no-deal Brexit.

If the new prime minister sought to avoid a new session, and prorogued immediately upon entering Downing Street at the end of July, the country would be without a parliament for three months. Since the 1980s the normal length of a prorogation has been two weeks or less. A minority government preventing parliament from scrutinising a decision opposed not only by MPs, but also, according to survey evidence, by the people, would appear an outrage. The ensuing demonstrations would make the People’s Vote march look like a tea party.

Is there any other way to break the deadlock? For most MPs, the main obstacle to the withdrawal agreement is the Irish backstop. That either requires Westminster to replicate all EU tariffs and regulations, making Brexit pointless, or if Westminster diverges, it requires Northern Ireland to retain EU tariffs and regulations to remain congruent with the Irish Republic and avoid a hard border on the island of Ireland. The consequence, however, would effectively be a border in the Irish Sea and a weakening of the union, with Northern Ireland becoming detached from the rest of the UK.

It is understandable if Tory Brexiteers, who hoped that leaving the EU would mean “taking back control”, are unwilling to be impaled on this Morton’s Fork. There is no chance of the withdrawal agreement being accepted by the Commons unless the backstop is amended.

The EU has declared that an amendment is not possible. But, though rigid in theory, the EU is often flexible in practice. It has indeed already shown flexibility on the backstop – for the withdrawal agreement allows Northern Ireland to remain in the EU internal market without accepting more than a fraction of the regulations of that market. The EU, with the consent of Ireland, now needs to show further flexibility by providing for the backstop either to be unilaterally terminated by Westminster, or, alternatively, to allow a sunset clause to be attached to it so that it automatically comes to an end after a certain period of time.

‘A no-deal Brexit would be worse for all concerned, and especially for Ireland.’ Photograph: Neil Hall/EPA

The danger of such an amendment, some might object, is that an extreme Brexiteer such as Raab, freed of the backstop, would allow a hard border in Ireland. This would go against the spirit, though not the letter, of the Good Friday agreement.

But this danger is more apparent than real. The vast majority of MPs are determined to avoid a hard border, and the extreme Brexiteers, as well as the Brexit party, would be disarmed once parliament had passed an amended withdrawal agreement.

Of course, EU agreement to an amended backstop must depend upon the consent of the Irish government. But if that consent is refused, and the EU remains adamant, Tory MPs might conclude that the EU has been unnecessarily hostile. In those circumstances, their hostility to a no-deal Brexit might well evaporate, as it in effect did yesterday. The only alternative, given Tory opposition to a further referendum, would be a general election, leading to huge gains for the Brexit party and a possible Corbyn government.

But why, in any case, should the EU grant a further extension? The only argument for it would be that MPs had not been able to make up their minds. That may not prove very persuasive to the leaders of the other EU member states over three years after the referendum. Nor is it at all clear how a further extension could break the deadlock.

A no-deal Brexit would be worse for all concerned, and especially for Ireland – which would suffer both from a hard border and from damage to its economy. Generosity, therefore, on the part of the Irish government would, not for the first time in Anglo-Irish relations, be more sensible than intransigence; and it would help renew the good relations between Britain and Ireland – relations that have been so badly damaged by the Brexit process.

• Vernon Bogdanor is professor of government at King’s College, London, and author of Beyond Brexit