Labour MP Kim Leadbeater, who introduced the bill in the Commons, told the BBC she was a “mixture of feeling extremely disappointed and upset and also quite angry” that the legislation would fall in the Lords.
She said there were MPs willing to take the same bill on during the next session should they be successful in the private members’ bill ballot, which can guarantee debating time on Friday sittings.
Leadbeater said she hoped it would then clear the Commons again and agreement could be reached with peers over amendments.
She also acknowledged the powers in the Parliament Acts could be used to prevent it falling again.
Under the terms of the rarely used legislation, if an identical bill passes the Commons a second time then the Lords cannot prevent it progressing again and it would become law at the end of that second session even without peers’ approval.
The Parliament Acts were last used in 2004 to push through a ban on fox hunting.
Leadbeater said: “We don’t want to get to that stage, we want this to go through the normal legislative process.”
Baroness Grey-Thompson, who opposes the bill, said peers had been scrutinising the bill line by line and it was a “red herring” to criticise the number of amendments tabled.
She told the BBC: “When it fails the biggest reason is because it’s a poorly written bill.
“It was written in haste and there are so many gaps in it that a number of peers are really uncomfortable with this particular bill, even though they may be in favour of the principle.”
Baroness Grey-Thompson said she was “not against the principle”, adding: “But if we’re going to do this, we have to have safeguards and I really don’t think there are anywhere near enough safeguards in it.”
Labour peer Lord Falconer of Thoroton, the bill’s sponsor in the Lords, has tabled a proposal which would allow peers to discuss the issue more generally on Friday rather than continue making progress through the amendment paper.