In its report, the National Audit Office said that political uncertainty and delays in negotiations with the European Union have hampered preparations for new border arrangements, and the government is now racing to bolster computer systems, increase staffing and build new infrastructure to track goods.
"Without data-sharing, our exports will grind to a halt", he said.
At Tuesday's meeting, Geoffrey Cox reportedly told May that an indefinite backstop would be like being stuck in Dante's first circle of hell, while Jeremy Hunt also insisted that the United Kingdom should have the right to withdraw unilaterally from the mechanism.
The reports said: "Government's assumption that the risks will not change materially on day one is reasonable in the short term but organised criminals and others are likely to be quick to exploit any perceived weaknesses or gaps in the enforcement regime".
Such a situation, coupled with a possible loss of access to European Union security and law enforcement resources, "could create security weaknesses which the government would need to address urgently", the report said.
According to people familiar with the matter, seven senior pro-Brexit ministers spoke out against a proposal that could allow the United Kingdom to stay inside the European Union's tariff regime indefinitely.
Responding to the leak, a Downing Street spokesman said: "This is nothing more than a partial reflection of advice to ministers, and not of decisions taken - the Prime Minister made her position absolutely clear in the House of Commons on Monday".
Fears that the United Kingdom is heading for a no-deal Brexit have dramatically escalated after ministers warned the country could plunge into economic chaos and even need to commission ships to import food and medicines.
The Department for Transport (DfT) is known to be looking at options for freight if major routes, such as Dover to Calais, are clogged with vehicles held up by customs delays.
Johnny Mercer, who earlier this week branded the Government a "shitshow", said he had received "overwhelming" support from within the party after speaking out. It also identified two "uniquely difficult" post-Brexit challenges: arrangements for freight trucks carried on so-called roll-on, roll-off (ro-ro) ferries, such as those between Dover and Calais; and the situation at the Ireland-Northern Ireland border.
Chartered ships would be used to bring goods into other ports.
Declan Billington of the Northern Ireland Food & Drink Association said that the "viability of the dairy section is being brought into question" because 30% of milk from Northern Ireland is processed in the Republic of Ireland.
On no-deal planning, the report details the extra responsibilities that will be placed on both businesses and officials, saying that such a departure would result in the current annual total of 55m customs declarations needed for the UK's borders rising to 260m.