In one of his first meetings with House Speaker Kevin McCarthy (R-Calif.) after the recent debt ceiling fight, Rep. David Joyce (R-Ohio) knew he had to get a point across.

He broke out his “OK” pin, one of a slew he had distributed in the heat of January’s speakership fight to members of the Republican Governance Group, the ideological faction he chairs made up of moderate and pragmatic lawmakers. “OK” means “Only Kevin” ― a not-so-subtle reminder of where his group’s allegiance lies as other colleagues test the boundaries of McCarthy’s speakership. McCarthy spotted the pin and greeted Joyce with a laugh.

Joyce’s gesture was meant to offer some levity, but it held a deeper significance: It was a reminder to leaders not to forget the scores of Republicans who are standing by leadership after dealing with the theatrics of a small group of far-right colleagues. If leadership is going to negotiate with the far-right, they should also consider the asks from their loyal flank.

A small group of staunch conservatives last month blocked any bills from being considered on the House floor for more than a week, a rebuke to McCarthy over a debt ceiling bill they thought was subpar. That blockade was followed by a string of votes, forced by House Freedom Caucus members, on red-meat issues to satiate the base.

“The speaker did nothing wrong,” Joyce said in defense of McCarthy cutting a deal with Democrats on debt ceiling legislation, which exacerbated the schism in the conference. “So why are we making any concessions to anyone because this is how this place should work?”

The House’s focus on the far-right’s demands over the past month has irritated Republicans who represent swing districts or are worried that an extreme legislative agenda will push voters away and hand the House majority to Democrats in 2024. So they are learning to flex their procedural muscles, largely behind the scenes, to keep some proposals they see as most damaging off the House floor.

In recent weeks, these lawmakers have kept some abortion-related measures from being put to a vote and sunk an amendment that would have derailed a government oversight bill. They also have tried to convince their far-right counterparts to avoid altering appropriation bills during committee markups, warning that any poison pills could force a big enough group to reject the bills on the House floor if they feel they could hurt their reelection chances.

Several lawmakers who represent districts Joe Biden won have also asked leadership to go a step further and allow them in the negotiating room with their far-right colleagues during high-profile debates to explain why the groups’ demands could jeopardize their five-vote majority, according to two people familiar with the request who, like others who spoke to The Post, did so on the condition of anonymity to detail private conversations.

“We have the majority because we all created the relationship with our districts to earn the right to be here, which, by the way, means that we all have the right to have our voices and our constituents’ voices heard,” said Rep. Marcus J. Molinaro (R-N.Y.), a freshman representing a district Biden won. “There’s a lot more effort among the pragmatic members, if you will, to be sure that we’re talking more effectively with the Freedom Caucus members and that we’re all having this dialogue.”

Freshman lawmakers in particular have begun to put leadership on notice, asking in a recent meeting that the National Republican Congressional Committee not use any money freshmen raised on their far-right colleagues’ campaigns, according to two people familiar with the request.

Swing-district Republicans also have been pushing leadership to be strategic about which messaging bills they bring to the floor, arguing it’s not worth forcing vulnerable members to take tough votes on legislation that will die in the Senate. The roughly one dozen lawmakers interviewed for this story were all clear that Republicans’ main priority should be passing bills that address their constituents’ economic concerns.

“If we don’t go back to the reason that we were elected, we won’t be here for very long, because if people don’t feel at home that you’re getting what you promised or what they thought you were listening to, they’re going to go start looking for somebody else,” said freshman Rep. Lori Chavez-DeRemer (R-Ore.), who flipped her seat from Democrats in 2022.

While far-right lawmakers have mastered the art of drawing attention to their demands, others have purposely tried to influence leadership behind the scenes. Though they have no master plan to ensure their concerns are prioritized, they have found ways to band together to show that, simply by the numbers, certain measures will fail if brought to the full House.

The New York delegation has worked Republicans’ slim majority to their benefit after flipping four seats held by Democrats in the midterms. Often referring to themselves as part of the “majority-makers,” Long Island Republicans — excluding embattled Rep. George Santos — recently persuaded leadership to pull anti-union amendments from legislation, including an amendment that would have weakened collective bargaining rights.

Union members have historically supported Democratic candidates by wide margins, and they remain a key constituency targeted by Biden in his reelection effort. But Republicans have made inroads with labor groups in recent years as they gained a larger share of White working-class voters’ support.

“We’ve made it very clear that voting against organized labor is not something that we intend to do, and I think we’ve done a good job of taking a couple of amendments off the floor,” said Rep. Anthony D’Esposito, who is a former New York Police Department detective and described himself as a “pro-labor Republican.”

A group of Republicans also privately vowed to derail the Reins Act — a bill popular among the conference because it would provide more congressional oversight on agency regulations that affect the economy by $100 million or more annually — if an abortion amendment offered by Rep. Bob Good (R-Va.) had been adopted. Instead, 10 lawmakers voted down the amendment, which would have required congressional approval on any federal decision that expands abortion access, according to two people familiar with the dynamics.

Abortion and other policy issues have become a major point of contention between moderates and the far-right.

During a previously planned meeting in the office of Republican Whip Tom Emmer (R-Minn.) office last month that included Majority Leader Steve Scalise (R-La.), a group of lawmakers balked when Scalise said he’d put a bill on the floor that would permanently codify the Hyde Amendment, which bars taxpayer dollars from being used for abortions — a demand of the far-right faction that conservatives largely support. Scalise had initially included the legislation in a list of priorities that would pass the first week of the year. But after the meeting, leaders were forced to drop the effort for a second time because it would affect people who purchase insurance coverage through the Affordable Care Act marketplace.

A group of swing-district Republicans has vowed they will continue to defeat abortion-related amendments that come to the House floor, a person familiar with the discussion said.

But the ongoing appropriations process will be more complicated for those Republicans, as their more conservative counterparts are expected to tuck several strict antiabortion measures into must-pass spending bills. Those additions will not only create conflict within the House GOP but also with the Democratic-controlled Senate. A House subcommittee overseeing the Food and Drug Administration has already approved language limiting access to mifepristone, a prescription abortion pill — a limitation not all Republican lawmakers support.

A smaller group of lawmakers on the Appropriations Committee has also tried to prevent certain amendments from being attached to bills before they arrive on the floor. In a pre-markup meeting last month, Republicans on the panel ran through amendments to discuss and debate, including one from Rep. Andrew S. Clyde (R-Ga.) that would have struck a provision allowing for H2A work visas to be used on a year-round, rather than a temporary, basis.

Committee members Rep. David G. Valadao (R-Calif.) and Rep. Dan Newhouse (R-Wash.), who both represent rural districts, argued that they had never heard from a single farmer who believed limiting visas is a good idea, because those farmers need more workers to keep their agricultural businesses running. Clyde still brought up the amendment, which failed.

The swing-district Republicans are not challenged only by the right, but the left, too.

Valadao split with his party last month to vote for a Democratic amendment that would have codified giving a pathway to citizenship for undocumented migrants brought to the U.S. as children by their parents had it been adopted. Democrats will continue to find opportunities to divide Republicans on the issues and put vulnerable Republicans on the record to target them in their elections.

“We’re always put in these difficult positions,” said Valadao, who represents a district Biden won, and that Valadao lost in 2018 before winning it back in 2020. “Sometimes we’re making the base of our party unhappy because we’re not following exactly what they want us to do, but when you look at what the average American, and especially those of us who need independent voters, we are usually closer in line with what they want us to do.”

The appropriations process has also inspired lawmakers to remind their Freedom Caucus colleagues that demands for significant spending cuts will be passed through the House, but ultimately will be rebuked by the Senate. The upper chamber is controlled by Democrats, but even GOP appropriators there pledged to mark up spending bills based on the parameters set in the debt ceiling deal reached by the White House and McCarthy.

A growing fear among many Republicans is that Freedom Caucus members will force a government shutdown over spending cuts — a position some have publicly taken. While pragmatic lawmakers would vote to fund the government, reopening it might mean relying on Democrats again, a position that irked the far-right during the debt ceiling fight and caused them to again go after McCarthy.

Preliminary conversations on ways to circumvent what they say is far-right obstruction on the House floor have already taken place among some Republicans.

“If it continues, we’re going to have to come up with a different role with Democrats,” Rep. Don Bacon (R-Neb.) said. “We’re going to have to work around these guys,” he said of his Republican colleagues who have held up floor activity.

Even so, members of the Freedom Caucus do not appear likely to relent on their demands to quickly impeach Biden and some of his Cabinet officials. Vulnerable lawmakers have privately raised concerns that these efforts are a distraction and deviate from their constituents’ wishes for them to work on implementing policy.

In particular, moderates hope two resolutions that would expunge Donald Trump’s two impeachments, introduced by Rep. Marjorie Taylor Greene (R-Ga.) and Republican Caucus Chair Elise Stefanik (R-N.Y.), do not get a floor vote. If they do, they would force freshmen who never took part in his impeachments — and Valadao and Newhouse, who voted in favor of Trump’s impeachment after Jan. 6 — into perilous positions.

Greene dismissed moderate colleagues’ concerns, noting that a reason Republicans have the majority is because Trump’s base helped elect swing-district lawmakers.

“You know how they got elected?” Greene said referencing her swing-district colleagues. “They had Republicans that feel just like the base across the country that voted for them. They work for those members, too. They don’t just work for independent voters in their district.”

The sentiment that all Republicans must fall in line drives Joyce “nuts” and has caused him to directly cross a Freedom Caucus member last month who had publicly rebuked his colleagues for failing to side with the more conservative demands of the conference.

“At the end of the day, if you don’t want to vote for it, that’s fine,” Joyce recalled telling a Freedom Caucus lawmaker. “But do not get up and say that no Republicans should vote for this. There are 435 districts. You don’t control any of them but the one you were elected in.

“We’re not actively trying to side with Democrats. We’re just trying to do what’s right for our district.”

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