For the House of Lords, the unelected counterpart to the House of Commons, Wednesday could mark a rare moment in Britain’s politics: The ermine-robed barons and baronesses of that ancient chamber will vote on whether to defy an elected British prime minister over a flagship policy.

The Lords are scheduled to hold a pivotal debate on the policy, which would put asylum seekers on one-way flights to Rwanda. They have attached multiple amendments to the bill in an attempt to water it down; the government, with its hefty Conservative majority in the Commons, has systematically stripped them off.

Nobody, least of all the Lords themselves, believes that the upper chamber will ultimately torpedo the legislation. In the unequal clash between the elected Commons and the unelected Lords, the Lords invariably yield. But they could delay its passage by another week or two, which could be enough to jeopardize Prime Minister Rishi Sunak’s goal of putting the first flight to Rwanda in the air by the end of May.

That would thrust the House of Lords into election-year politics in Britain in a way that is unusual for an institution that views itself as a coolheaded, deliberative check on the more unruly Commons.

The prospect of sending asylum seekers to an East African country — and overruling a Supreme Court judgment — has generated so much opposition, even from Conservative peers, that it has shaken the Lords out of their customary deference.

“This is about people having a fundamental objection to a piece of government legislation,” said Simon McDonald, a former head of the British diplomatic service who became a cross-bench, or nonpartisan, member of the House of Lords, where he is known as Baron McDonald of Salford, in 2021.

“Personally, I would be disappointed if we just gave in,” he said. “For me, we need to go hard on conditions that have to be met before the act is put into force.”

Rwanda’s government, Mr. McDonald said, needed to demonstrate that it had put safeguards in place to ensure that the rights of asylum seekers who arrived there from Britain were not violated. Several of the Lords’ amendments are designed to do that, but the government has rejected them on the grounds that they are simply another legal hurdle to block the flights from starting.

For the government, timing matters. Mr. Sunak has championed the Rwanda policy as the best way to deter migrants who make the perilous crossing of the English Channel in small boats. Under the legislation, they would stay in the African nation even if they won refugee status.

Stopping those Channel crossings is one of his government’s five bedrock goals, and Mr. Sunak hopes the flights will help the Conservatives close a yawning gap in opinion polls with the opposition Labour Party.

But the policy has run headlong into concerns about human rights and the rule of law, which have agitated the normally equable Lords. The Supreme Court ruled in November that Rwanda was not a safe country for refugees, prompting the government to retool the policy to address those concerns — insufficiently, in the view of critics.

Several of the chamber’s members are retired judges and civil servants who view themselves as custodians of the courts and Britain’s adherence to international law. They are using the levers they have to force the government to remedy the legislation.

“The way the Lords operates, like so much of the British Constitution, is by convention rather than by rules,” said Richard Newby, the leader of the Liberal Democratic Party in the House of Lords. “The question is how far you push a convention rather than whether you break a rule.”

Mr. Newby predicted Mr. Sunak’s conservative government would not muster the votes of enough members on Wednesday to force the Lords to back down on the amendments. That means the bill would be kicked back to the Commons, most likely with fewer amendments.

The resulting back-and-forth could prevent the bill from becoming law until after the Easter holiday. Mr. Sunak has appealed to the Lords not to “frustrate the will of the people,” although recent polling suggests a majority of the British public do not support the policy.

The largest legislative assembly outside China, the House of Lords has some 800 members, including 91 who inherited titles, and 26 archbishops and bishops. Its ranks include former politicians, advisers and diplomats; most are appointed for life.

The Lords meet in an ornate chamber that, on busy days, has too few seats. Among them stands a gilded throne, inset with rock crystals and upholstered in red velvet, from which King Charles III speaks when he opens Parliament.

Members, who can shape laws and ask questions without the inconvenience of running for office, can claim up to 342 pounds, or $435, as a daily allowance. There are other perks, too: a desk in the Parliament complex; a parking lot; and plush, subsidized places to eat and drink, including the wood-paneled Bishops’ Bar.

But the members work, too.

“The Lords is the place where you get effective scrutiny,” said Jill Rutter, a senior research fellow at U.K. in a Changing Europe, a research institute. “The Commons gave the Rwanda bill virtually no scrutiny because it went through very quickly.”

“The problem,” she added, “is that the Lords basically knows it’s a ludicrous and illegitimate institution, which is why it almost always caves.”

Still, even within these constraints, the chamber can influence and even change policy. In 2015, the Lords persuaded the government to rethink cuts to welfare payments. Only last week, it was the prospect of defeat over an amendment to a bill in the Lords that prompted the government to pledge new rules barring foreign state ownership of British newspapers and magazines.

David Lipsey, a Labour member of the Lords, said he expected his party to press for about half a dozen amendments. He said it was “pretty unlikely” that Labour would keep up its opposition after Wednesday, even though there was a case for doing so.

“The Lords has always had the backstop role in stopping governments doing things that are beyond the bounds of democratic and legal decision-making,” said Mr. Lipsey, who became a member in 1999 as Baron Lipsey of Tooting Bec.

While Labour’s double-digit polling lead means it is likely to form the next government, the party’s leaders know that, if elected, they would not have an automatic majority in a chamber where many members are nonaligned.

“Labour doesn’t particularly want to establish the precedent that it’s OK for the Lords to chuck out a flagship piece of government legislation because there may be things they want to do,” Ms. Rutter said.

The election, most likely set for this fall, has given nonaligned members pause as well. Some worry about being painted as obstructionists by the government, which could seize on the unelected body as a weapon in a campaign. Others worry about constitutional reforms that could threaten their status.

The archbishop of Canterbury, Justin Welby, spoke out strongly against an earlier version of the Rwanda bill last year, saying it “fails to live up to our history, our moral responsibility and our political and international interests.”

But in an interview last December, he said, “I would like to play as small a role as possible in the debate. We’re within a year of an election.”