Amid fierce cabinet squabbling over the minutiae of customs policy, it is easy to lose sight of the big picture: a battle to decide what kind of country the UK will be after Brexit.

On the face of it, Wednesday’s Brexit sub-committee meeting provided little in the way of enlightenment. Downing Street once again deferred a decision over which of two customs proposals the UK will take into the next phase of negotiations with the EU.

Even if the cabinet does eventually decide between the hi-tech “maximum facilitation” model or the highly integrated new customs partnership (NCP) approach, the EU remains deeply sceptical that either will prove satisfactory.

But postponing the painful choice, or finding another way to fudge it, will no longer be enough for Theresa May. The one lasting legacy of Wednesday’s focus on customs has been to cast doubt on her government’s overall compromise approach to Brexit. What the Europeans have dismissed as the UK’s “have-cake-and-eat-it” fantasy is now under withering attack by Tory Eurosceptics too.

The target of their attack is the NCP approach still favoured by May and the Treasury. The idea of this plan, first outlined in August, is that post-Brexit Britain can avoid introducing friction at the EU border while at the same time creating enough flexibility to strike new trade deals with other countries.

It would do this by acting as if Britain were still in the existing customs union, but crucially allowing imports bound purely for UK customers to qualify for a rebate if there was a gap between the EU tariff rate and any reduction agreed under new trade deals.

Americans cars arriving at Southampton, for example, would still be subject to the 10% EU import duty, but UK showrooms might claw this back if the government succeeded in striking its desired new trade deal with the US.

More importantly, British car exports sent to Europe via Dover would not be held up by having to prove where their component parts came from – the so-called rules of origin test – because everything from outside Europe would have passed through the same common external tariff regime.

The problem is that delays at customs are not just caused by tariffs and rules of origin tests. A big factor in smoothing trade flows is standardising product safety standards and other regulations too. To maintain truly frictionless trade with Europe the UK would also need to abide by the rules of the single market.

Where the critics agree is that this would be impossible to do without effectively being a member of the single market and the customs union.

Jacob Rees-Mogg’s European Research Group (ERG) regards this as a sellout and a betrayal of the referendum result. The EU views anything less than explicit membership as an attempt at cherry picking and would force the UK to submit to the supervision of its courts and regulators too.

In fact, the more scrutiny it comes under, the closer the NCP looks to the partial customs union approach now favoured by Labour and a number of backbench Tory rebels. Instead of providing a halfway house a customs partnership could quickly prove to be the very softest of Brexits.

The only alternative would be the “highly streamlined” technological approach designed to provide “maximum facilitation” of trade, but not promising to avoid controls entirely. The ERG says this is a price worth paying to maintain freedom from EU rules and a more unfettered ability to strike independent trade deals.

The cabinet hardliners argue that if this proves unsatisfactory to the Irish government and collapses the talks, so be it – it would be the EU choosing to erect border controls, not the UK. “No deal is better than a bad deal,” as the prime minister used to say.

They say the real choice is between a lightly regulated open economy trading largely outside Europe, and maintaining a highly integrated European-style social market that has to keep collaborating with Brussels. After Wednesday’s clash, pretending Britain can have both is an option May no longer enjoys.