The idea of the European Union is under threat as never before, as Marine Le Pen’s anti-EU campaign powers on, Hungary challenges first principles and Britain lurches nearer the exit. What good timing for the Victoria and Albert Museum this week to open its redesigned European galleries, a strong reminder of our common history, told in artefacts. See how cultural influences from Italy and France ricochet across the continent; how trade spreads not just design and technology but ideas, as the Enlightenment breezes across borders. “We have a shared inheritance. Whatever they say, they can’t take that away from us,” says Martin Roth, the museum’s German director.

Querulous British negotiations look tawdry against that sweeping backdrop. As he celebrates 10 years as party leader, David Cameron is stepping closer to taking Britain out of the EU: will he make such a momentous move for such trivial reasons? Caught in a trap of his own devising, he has twisted together two wicked political issues – Brexit and benefits – like a pair of snakes.

His recent fruitless talks with EU leaders leave the British question somewhere between low and nowhere on the agenda for the meeting of the EU’s European council next week. The other 27 members have life-and-death matters to discuss: an unstoppable flow of migrants, terrorists crossing borders and Schengen crumbling, while the euro still trembles unsteadily. Patience with the frivolity of Britain’s pointless negotiation has worn thin.

For the first time observers are reporting that Brussels and Berlin regard a British exit as a serious possibility. They don’t want us to go, but Cameron’s cavalier approach pushes them towards indifference. Some report that he himself seems to care less about leaving than he once did, as UK polls sway closer to exit – and he hits a brick wall through his own ineptitude.

Why did he make the emotive but empty issue of abolishing working benefits for EU migrants the centre-piece of his renegotiation? His officials and government lawyers warned this was an impossible demand. Other leaders will not, and cannot, in law agree: free movement of labour without discrimination is a founding EU principle.

Besides, leaders in Poland, Hungary and other states must resist discrimination against their citizens or risk their own political survival. Why did Cameron ignore legal advice? He seized on a report by Open Europe, a sceptical thinktank, that said denying benefits to EU migrants is legal. But the law is not so straightforward, and now he’s in a fix.

At first it looked as if a simple residency test might do the trick: people qualify for in-work benefits only after living here for four years, so our own 18-year-olds would automatically qualify. But EU lawyers say, rightly, that this would discriminate against new arrivals who couldn’t have lived here from the age of 14. The only legal answer is to deny tax credits and housing benefit to everyone – including Britons – who hasn’t contributed by working for four years. When Iain Duncan Smith, the work and pensions secretary, was asked on yesterday’s Andrew Marr Show to rule that out, he refused – and praised the contributory principle that only pays out to those who have paid in.

How many people would be hit? The usual thinktanks can’t produce exact figures yet, but here are their informed calculations: 24,000 people aged under 20 draw tax credits – young parents unlikely to have four-year work histories; more are out of work; and a further 100,000 people aged 20 to 24 have been in work for less than four years. The total affected would be around 200,000, mostly parents, who draw significant tax credits.

Here’s the irony: that’s many more than the number of EU workers who draw these benefits.

Others who would be harmed – this might alarm Tory MPs – are stay-at-home mothers who have never worked. A third of marriages end in divorce, and mothers suddenly caring for children alone would be denied tax credits and housing benefit for four years as they work for the first time.

Wouldn’t this cause the same furore George Osborne created with his threatened tax credit cuts? Tory advocates of this solution to the EU conundrum say a four-year delay would be more politically acceptable as it would deny benefits to new claimants rather than taking cash from existing ones. After all, the tax credit U-turn merely shifted the cuts to new claimants of universal credit: so long as you don’t snatch benefits from any current recipients, it seems no one notices.

But there’s another wicked twist in all this. After Duncan Smith had sounded somewhat sanguine on the Marr show about the possibility of imposing a four-year wait on UK citizens, sources close to him suggested something very different: no, he would be strongly opposed as it would be so unfair, depriving too many young people and families of benefits they need.

Really? This was a surprising softening of the heart from the man who has overseen colossal benefit cuts and the bedroom tax – with another £12bn of cuts to come. As the Child Poverty Action Group protests, his welfare reform bill gives up poverty targets and stops measuring it altogether. So why the sudden attack of empathy?

Here is where the two snakes intertwine. Remember Duncan Smith is one of John Major’s original anti-EU “bastards”, and sits on the cabinet sub-committee overseeing EU negotiations. By refusing to let UK citizens suffer this four-year benefit delay, might he be ushering the country towards the Brexit gate he’s always favoured? If he is sacked or resigns for not accepting the renegotiated conditions, how principled he can sound.

If you need another reminder of just how crazed the Leave campaign is, consider its response this week to rumours that Cameron may campaign for “out” if he can’t win his impossible benefit demand. The Brexits say they don’t want him to lead them: he’s toxic, and they prefer Boris Johnson as their figurehead. Their perversity knows no bounds; the prime minister’s weight behind them would surely swing the referendum their way.

Britain’s future hangs on the short-term political whim of this prime minister. Stroll through the V&A’s gallery and contemplate how deep our roots lie in this common European culture.