Downing St insists Britain will quit the customs union despite several reports claiming Mrs May is ready to compromise with Brussels on the issue. One of May's senior aides told a meeting last month that her team would "not be crying into our beer" if Parliament forces the Government's hand.
While they still need approval in the House of Commons, the changes add to the difficulties facing May by showing the strength of opposition for her Brexit vision.
The EU Withdrawal Bill debate will inform the biggest battle over Brexit, expected in October or November, when lawmakers vote on the final deal negotiated with Brussels.
MPs vote on a motion to compel the government to drop its policy on the customs union later this week.
The Prime Minister "is trapped between demands from the Brexiteers who won the referendum, and the numerical reality of the House of Commons and what she can get through", says The Times's Matt Chorley.
"The UK will leave the European Union as a whole, all parts of it and that in leaving we must leave all constituent parts of the EU".
In the main reverse, peers backed a cross-party move to retain key EU human rights provisions on exiting the union.
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. This means the United Kingdom is not allowed to broker its own trade deals.
The Government is negotiating what Britain's customs arrangement will be like post-Brexit.
Rees-Mogg believes that after leaving the union, the United Kingdom should phase out all tariffs, in order to reduce consumer prices and stimulate competition.
Lord Keen warned that to incorporate the whole of the charter into United Kingdom law would do "serious damage" to the constitutional settlement post-Brexit.
Mr Umunna said on Twitter: 'The Brextremists threaten May with a leadership election if she concedes on the customs union but a new Tory leader would face the same parliamentary arithmetic'.
"There seems zero chance of having infrastructure in place by 2019 to deal with the customs checks that would be required", the broadcaster adds. They argue this is the only way of avoiding a hard border between Northern Ireland and the Republic.
"Turkey has a customs union with the EU but isn't in the bloc or the single market", the news website notes.