Last week's defeat, the first of two on that day, increased pressure on May to reconsider her refusal to remain in a customs union with the EU.
The battle over customs rules is likely to reach a climax in a Commons vote as early as next month.
But business leaders argue that leaving the customs union will severely damage cross-border trade, potentially result in new tariffs, and require burdensome checks on the origin of goods in transit.
Leaving the customs union would mean Britain can negotiate its own free trade deals with non-EU countries. "So we want to come out of the customs union".
Lord Keen said that to suggest acts of parliament should be capable of being "struck down by reference to a body of foreign law" would be "one of the greatest constitutional outrages since 1689".
But the prime minister's spokesman also ruled out abandoning one of the two customs options Britain has proposed to the European Union, pointing out that both had been included in her Mansion House speech, which was signed off in advance by the entire cabinet.
Remainer MPs hope that Irish and EU negotiators will hold out for a solution to the Border issue that keeps the whole of the United Kingdom in a customs union.
In the main reverse, peers backed a cross-party move to retain key EU human rights provisions on exiting the union. This means the United Kingdom is not allowed to broker its own trade deals.
Those in favour of remaining in the customs union argue that cutting trading ties would severely damage the United Kingdom economy.
Crucially, keeping the customs union would also "go some way to resolving the thorny question of the Irish border, and would allow the United Kingdom to keep its red line of controlling immigration", says Bloomberg.
Earlier this year, a leaked government report predicted that leaving the customs union would reduce the growth of the United Kingdom economy by up to 8% over the next 15 years.
May's aides said on Sunday that the key policy on Brexit hasn't changed and insisted she's committed to breaking away from European customs laws.
They have forced themselves into a trilemma in which it has adopted three positions, only two of which can be achieved at any one time - to avoid a hard border within the island of Ireland, for the United Kingdom as whole to leave both the customs union and the single market, and to rule out any special arrangements for Northern Ireland in relation to a customs union and single market.
"Turkey has a customs union with the EU but isn't in the bloc or the single market", the news website notes.