Days before an all-but-certain government shutdown instigated by hard-right Republicans in Washington, Elizabeth Catalino arrived at a Hudson Valley high school looking for answers from her own G.O.P. congressman, Representative Mike Lawler.

“I’m very disappointed that you have this band of Republicans that are being so obstinate,” Ms. Catalino, who described herself as an independent, said in an interview on Monday after she attended a town hall-style meeting with Mr. Lawler in East Fishkill, N.Y. “And I think it’s just going to be hurtful to everybody if they’re successful in shutting the government down.”

As right-wing lawmakers take Congress to the brink of a government closure for which their party would almost certainly bear the blame, dozens of Republicans — particularly those like Mr. Lawler who represent districts won by President Biden — are toiling to head off the backlash from voters for the chaos sown by some of the most extreme members of the G.O.P.

“Especially in a divided government, we’re going to have to find compromise. We’re going to have to find areas of agreement,” Mr. Lawler said in an interview. “And for the handful of people that are unwilling to do that, it’s frankly destructive to the country and really harmful to the American people.”

Mr. Lawler is one of 18 House Republicans representing a district that voted for Mr. Biden, meaning that he must appeal to constituents ranging from supporters of former President Donald J. Trump to independents like Ms. Catalino and centrist Democrats. Before him, the seat had not been held by a Republican for more than two decades. And Mr. Lawler is expected to face a tough re-election race in 2024 as Democrats aim to wrench the seat back.

At the town hall and on cable news shows over the past week, Mr. Lawler has emphasized his opposition to a shutdown and his efforts to bring a bipartisan spending patch to the floor. He has told his swing-district constituents that the government is hurtling toward a closure at midnight on Saturday because a handful of G.O.P. hard-liners have stood in the way of government funding while he and other Republicans were working to extend it.

“Apparently, he’s willing to work with other people and other parties,” Eric Eckley, another of those constituents, said outside the town hall. “But that doesn’t answer the question of why the people in his party are not willing to run our government.”

Mr. Lawler’s office barred reporters from attending the meeting. But outside, attendees expressed frustration with the mess in Congress, even as some came away from the event with sympathy for their congressman’s plight.

“We’re all just frustrated,” Ms. Catalino said. “We’ve had enough. We’ve had enough of this bickering and this dysfunction in Washington.”

Democrats have already seized on the disorder to bash vulnerable Republicans like Mr. Lawler for a shutdown that has not yet begun. House Democrats’ campaign arm has sent out political attacks tying them to the most extreme members of the conference.

“Mike Lawler is once again blindly following his crew of extremist House Republicans, this time leading our country into a government shutdown that will devastate millions of Americans,” Dora Pekec, the New York press secretary for the House Majority PAC, the main outside group that funds advertising for congressional Democrats, said in a statement that echoed several others targeting similarly situated Republicans.

The Senate is debating a bipartisan agreement that would extend federal funding through Nov. 17 and provide billions in aid to Ukraine and disaster relief. It contains elements of a bipartisan framework put forward by the centrist Problem Solvers Caucus in the House, of which Mr. Lawler is a member. But with far-right lawmakers opposing any stopgap funding bill, neither of those proposals — nor a more conservative-leaning alternative proposed by Speaker Kevin McCarthy — has enough Republican votes to pass the House.

And Representative Matt Gaetz of Florida and other far-right Republicans opposing a temporary funding bill have vowed to try ousting Mr. McCarthy from his post if he works with Democrats to get one across the finish line.

That leaves mainstream Republicans who stand to pay the steepest political price frustrated.

“It’s Munchausen at work: creating a crisis, blaming others, pretending to fix it. And then rinse, wash, repeat,” Mr. Lawler said.

He and other vulnerable Republicans have explored working with Democrats to keep the government open. A bipartisan group introduced a bill Wednesday to stave off a shutdown, though it faces steep hurdles to receive a vote on the floor without Mr. McCarthy’s support.

“I am ready to explore each and every option possible to make sure that we don’t shut this government down,” Representative Anthony D’Esposito, Republican of New York, said on C-SPAN’s “Washington Journal” on Wednesday. “Every option is on the table.”

Representative Marc Molinaro, Republican of New York, also has been deeply involved in negotiations to prevent a shutdown.

“The goal here is to avert a shutdown and to intervene in federal spending,” Mr. Molinaro said, adding that his constituents have been split on whether their priority is cutting federal expenditures or keeping money flowing.

Mainstream Republicans have made clear their frustration with their hard-line counterparts, whose opposition to a stopgap measure has placed them in a challenging position.

“We’re giving up hundreds of billions of dollars for five guys that just want to go and run their Twitter accounts and do political fund-raising based on extreme positions,” Representative John Duarte, Republican of California, said last week after a handful of right-wing lawmakers blocked a bill to fund the military from being considered on the floor. “The controversy seems to be the goal.”

Republicans trying to keep the government funded have also warned that their party will take the blame for a shutdown, and that the White House will use that to its benefit.

Still, Mr. Lawler predicted that the hard-line Republicans blocking government funding would take the brunt of any political backlash, and some of his constituents seemed to agree.

“If he’s persistent enough, that’ll come through,” Anthony Scarrone, who has lived in East Fishkill for 38 years, said of Mr. Lawler. “I think people will see that. And that’s important.”