When the Prime Minister first assembled his cabinet, the most controversial appointment was Suella Braverman as Home Secretary. She had only just left the role under Liz Truss after she admitted sending an official document from a personal email account. But when Truss fell, Braverman called for Rishi Sunak rather than a Boris Johnson restoration. She was back in the Home Office after less than a week.

Some suspected a grubby deal between the two, but Sunak had plenty of reasons to want Braverman back. While critics accuse her of harbouring unsubtle leadership ambitions, her place in the cabinet keeps an important part of the Tory coalition on side – even if it comes with downsides. Every now and again, she’ll say something controversial that grabs attention. But for the quiet Sunak, this can be useful. It helps his party defend itself against attacks from the right.

This week, Braverman faced allegations of breaking the ministerial code again, this time by asking civil servants to investigate arranging a special one-on-one speed awareness course for her, rather than her attending in a group. That a cabinet career could be threatened on such a technicality was seen by many Tories as proof that the civil service is in open revolt, leaking and briefing against ministers and making the country ungovernable.

But the bigger problem for Sunak in the Home Office is not personnel – it’s policy. Figures this week show net migration has surged to more than double pre-Brexit figures. When the referendum campaign promised controlled immigration, almost everyone presumed this meant the number would fall. Instead, it’s on course for an historic high.

The government does not, of course, set the numbers. It sets the criteria for entry (salary level, type of job), then sees how many people come. The figures show that far more came last year than was anticipated. As Home Secretary, Theresa May tightened the criteria (for non-EU citizens) with the backing of George Osborne, then chancellor. Both David Cameron and Osborne regarded reducing immigration as a key pledge, displayed on posters at party conference. But now the Home Office is fighting a lonely battle: the Treasury is desperate for economic growth.

While a clampdown on visas for dependants of students has been announced, further constraints, such as raising the salary threshold for visas, are seen as unlikely – especially in the absence of a successful Ukip-style challenger talking about immigration. ‘There isn’t a serious threat to the right of us – at least for now,’ says a minister. ‘It means No. 10 are more worried about the economy.’ Yet the prospect of a voter backlash – even Labour now argues net migration is too high – is a cause for concern across the party.

For Sunak, legal migration is a conversation he would rather avoid for now, and he is refusing to put a figure on where it should be. ‘Stop the boats’ is his mantra and illegal migration his focus. Progress here could add to a sense of control that could make legal migration less contentious. ‘You have to stop the boats before doing anything major on legal migration,’ says one government aide. The problem is that, while Sunak already has the powers to cut legal migration, stopping the small boats depends on the courts.

The Prime Minister’s plan is to start deportations to Rwanda: he thinks this will reduce the number prepared to pay thousands to people smugglers. At present, the chances of deportation are slim, so for many the costs are seen as worthwhile. But if there is a serious risk of deportation, as is the case for Albanians (more than 1,000 have been returned since deportations started in December), the business model collapses. This, at least, is Sunak’s theory.

Johnson’s Rwanda deportation scheme was stopped by the European Court of Human Rights in Strasbourg, pending legal consideration. The government won the first ruling, but campaigners appealed. The Court of Appeal’s verdict is expected in the coming weeks. If it finds in favour of the government, campaigners are likely to go to the UK Supreme Court. It could refuse to consider the case, saying the Rwanda scheme has already been ruled lawful by very senior judges and does not merit yet another appeal. If so, deportations could start by September.

Campaigners would then go back to the ECHR, but this would not stop deportations. The blessing of three UK court rulings could be enough for the government to get Rwanda-bound planes on the runway, potentially with caveats such as a promise to return individuals if Strasbourg eventually found against them. By then, the Illegal Migration Bill may have passed into law, explicitly giving ministers the power to ignore Strasbourg and the interim junctions from the court, known as Rule 39 orders, that halt deportation flights. This would allow the government to get flights going on a significant scale.

If Strasbourg does somehow manage to frustrate the scheme or the UK courts find against the government and declare Rwanda is an unsafe country, this would mean a showdown. The Tories could pledge a referendum on leaving the ECHR in their manifesto, hoping to reconvene the Brexit alliance of voters that delivered the 2019 majority. (The assumption is that the public would support the Rwanda scheme by a margin of two to one.) ‘It’s either stop the boats or leave the ECHR,’ says one senior Tory.

Proponents argue there would be an emotive case to make. It would include other elements of the ECHR which can thwart UK government aims, such as Article 8 that can stop criminal deportations on the basis of the right to family life. One figure close to Sunak suggests that the PM could even campaign to leave the ECHR in the event of a referendum – casting it as a point of high principle, democracy and sovereignty. Unless, of course, the UK Supreme Court strikes down the Rwanda scheme and the Tories end up going to Strasbourg to ask its judges to overrule those in London. This option is being considered.

A referendum on whether to leave the ECHR would divide the Tory party. Attorney General Victoria Prentis is among those who have warned against leaving – the ECHR is written into the Good Friday Agreement and Brexit deal. But if immigration numbers stay high and Sunak can’t stop the boats, it may be the one lever he has left to pull.