ANCHORAGE — Sitting in a darkened exhibition room at the Anchorage Museum on a recent Tuesday morning, Senator Lisa Murkowski of Alaska conceded that she might lose her campaign for a fourth full term in Congress, where she is one of a tiny and dwindling group of Republicans still willing to buck her party.“I may be the last man standing. I may not be re-elected,” she said in an interview after an event here, just days after breaking with the G.O.P. to support confirming Judge Ketanji Brown Jackson, President Biden’s Supreme Court nominee and the first Black woman to serve there. “It may be that Alaskans say, ‘Nope, we want to go with an absolute, down-the-line, always, always, 100-percent, never-question, rubber-stamp Republican.’“And if they say that that’s the way that Alaska has gone — kind of the same direction that so many other parts of the country have gone — I have to accept that,” Ms. Murkowski continued. “But I’m going to give them the option.”In a year when control of Congress is at stake and the Republican Party is dominated by the reactionary right, Ms. Murkowski is attempting something almost unheard-of: running for re-election as a proud G.O.P. moderate willing to defy party orthodoxy.For Ms. Murkowski, 64, it amounts to a high-stakes bet that voters in the famously independent state of Alaska will reward a Republican centrist at a time of extreme partisanship.She has good reasons to hope they will. Though it leans conservative, Alaska is a fiercely individualistic state where the majority of voters do not align with either major political party. And under a new set of election rules engineered by her allies, Ms. Murkowski does not have to worry about a head-to-head contest with a more conservative opponent. Instead, she will compete in an Aug. 16 primary open to candidates of any political stripe, followed by a general election in which voters will rank the top four to emerge from the primary to determine a winner.Despite her penchant for defecting from the party line, Ms. Murkowski also has powerful help from the Republican establishment; Senator Mitch McConnell’s leadership political action committee announced last week that it had reserved $7.4 million worth of advertising in Alaska to support her candidacy.So she has embarked on a re-election campaign that is also an effort to salvage a version of the Republican Party that hardly exists anymore in Congress, as seasoned pragmatists retire or are chased out by right-wing hard-liners competing to take their places.“The easy thing would have been to just say, 20 years is good and honorable in the United States Senate. It’s time to, as I always say, it’s time to get my season ski pass at Alyeska and really get my money’s worth,” Ms. Murkowski said, referring to the nearby ski resort. “But there is a different sense of obligation that I am feeling now as a lawmaker.”Still, Ms. Murkowski, the daughter of a former Alaska senator and governor, faces a tough race. Her vote last year to convict former President Donald J. Trump at his impeachment trial on a charge of inciting the Jan. 6 attack on the Capitol prompted Alaska’s Republican Party to censure her and join Mr. Trump in embracing a right-wing primary challenger, Kelly Tshibaka.Ms. Murkowski is the daughter of a former Alaska senator and governor.Ash Adams for The New York TimesA view of Anchorage, the biggest city in a famously independent state.Ash Adams for The New York TimesAnd while there is now no Democrat going up against Ms. Murkowski in the race, it is not clear whether she can attract enough support from liberal voters to offset the conservatives who have been alienated by her stance against Mr. Trump. Many liberals have been angered by Ms. Murkowski’s opposition to sweeping climate change policies, as well as her support in 2017 for the $1.5 trillion Republican tax law that also allowed drilling in the Arctic National Wildlife Refuge.So Ms. Murkowski has been reminding voters of her flair for pursuing bipartisan initiatives, such as the $1 trillion infrastructure law that is expected to send more than $1 billion to her state, and promoting her strong relationships with Democrats. At an Arctic policy event in the Dena’ina Civic and Convention Center, she appeared with Senator Joe Manchin III, the centrist West Virginia Democrat, who was wearing an “I’m on Team Lisa” button and proclaimed, “I’m endorsing her 1,000 percent.”All of it is fodder for her staunchest opponents. Ms. Tshibaka, a Trump-endorsed former commissioner in the Alaska Department of Administration, has worked to paint Ms. Murkowski as a liberal and to rally the state’s conservative base against her. She is trying to capitalize on longstanding antipathy for the senator on the right, which was incensed when she voted in 2017 to preserve the Affordable Care Act and by her opposition in 2018 to Justice Brett M. Kavanaugh’s confirmation.“It’s time for a change. We feel forgotten,” Ms. Tshibaka told supporters at the opening of her Anchorage campaign office this month. “We feel unheard, and we don’t feel like these votes and decisions represent us.”Standing atop a desk, she urged them to “rank the red,” meaning to place her as their top choice without ranking any other candidate on the ballot.“We feel unheard, and we don’t feel like these votes and decisions represent us,” said Kelly Tshibaka, Ms. Murkowski’s leading challenger.Ash Adams for The New York TimesMs. Tshibaka, whose campaign did not respond to requests for an interview, told the crowd of supporters how Ms. Murkowski’s father, Frank, named his daughter to finish out his term as senator once he became governor in 2002, deriding what she called the “Murkowski monarchy.”Supporters grabbed slices of pizza and picked up bumper stickers, as well as decals that showed Ms. Murkowski embracing President Biden.“Nothing surprises me at this point. I don’t understand why she makes the decisions she makes,” said April Orth, 56, who called Ms. Murkowski’s vote to confirm Judge Jackson “an injustice to the people of the United States of America.”Ms. Tshibaka emphasized her conservative credentials and support from Mr. Trump, regaling the crowd with stories about her visit to the former president’s Mar-a-Lago resort in Palm Beach, Fla., for a campaign event. (The February event cost her campaign $14,477.10 for facility rental and catering, according to her latest campaign filing.)Joaquita Martin, 55, a paralegal, called Mr. Trump’s support “an incredibly powerful endorsement” of Ms. Tshibaka, adding that “I identify as a conservative, and Murkowski can call herself Republican all day long, but if that’s the definition of Republican, I’m out. That’s not me.”Ms. Murkowski’s decision to seek another term did not come lightly. Ms. Murkowski famously lost her Republican primary election in 2010 to a Tea Party-backed candidate, then ran anyway as an independent and triumphed in a historic write-in campaign with a coalition of centrists and Alaska Natives.April Orth, 56, is a supporter of Ms. Tshibaka. “Nothing surprises me at this point,” she said of Ms. Murkowski. “I don’t understand why she makes the decisions she makes.”Ash Adams for The New York TimesDeventia Townsend, a registered Democrat, and his wife, Charlene. “She has so much courage,” Mr. Townsend said of the senator. “She votes from her heart.”Ash Adams for The New York TimesOf the seven Senate Republicans who voted to convict Mr. Trump last year, Ms. Murkowski is the only one facing voters this year. She has not shied away from that distinction; she speaks openly of her disdain for Mr. Trump and his influence on her party. She has also supported Deb Haaland, Mr. Biden’s interior secretary and the first Native American to serve in the post, and boasted of her lead role in negotiating the infrastructure law.It has made for some unpleasant moments, she and her family say.“On one hand, had she chosen not to run, I would have been completely supportive because it’s just been like, ‘Damn girl, this has been a long haul,’” said Anne Gore, Ms. Murkowski’s cousin. “But on the other hand, you’re like, ‘Oh, sweet mother of Jesus, God on a bicycle — thank God you’re running’ because, you know, we can’t lose any more moderates.”While Ms. Murkowski has never secured more than 50 percent of the vote in a general election, this year she could stand to benefit from the new election rules, which advantage candidates with the broadest appeal in a state where most voters are unaffiliated.“I don’t think it changes their behavior, but it rewards behavior that is in line with the sentiment of all Alaskans, rather than the partisan few,” said Scott Kendall, a former legal counsel to Ms. Murkowski who remains involved with a super PAC supporting her re-election and championed the new rules.Mr. Kendall said his push for the statewide changes was independent from the senator’s campaign, arguing that his goal was “treating every Alaska voter the same and giving them the same amount of power.”There is little question that it has made for a friendlier landscape for Ms. Murkowski and appeals to the middle. At least one candidate, the libertarian Sean Thorne, jumped into the race because of the potential to prevail in a broad primary.For now, Ms. Murkowski is focusing on the basic needs of her state.Earlier this month, she stood, beaming, before about 1,200 local, tribal and community leaders who had flown across the state for a symposium explaining how Alaska stood to gain from the infrastructure law, which she singled out as perhaps her proudest accomplishment.“This is going to be an Alaska that is better cared for than ever before and an Alaska with a higher quality of life, whether you’re here in Anchorage or whether you’re in a remote village,” she declared. She mingled through the buzzing crowd, introducing herself as Lisa and embracing longtime friends.Ms. Murkowski’s campaign is focusing on the basic needs of her state and trumpeting the bipartisan infrastructure legislation that was passed last year.Ash Adams for The New York TimesTribal leaders talked about how the law would give them a chance to connect communities with broadband and ensure they had clean drinking water. A Kwethluk city employee waited to give the senator a handout describing a port project, while another village official asked for help with a broken washateria, first built in 1975, that had left them without running water since Christmas. And then there were the constituents who wanted a brief word about Ms. Murkowski’s work in Washington.Deventia Townsend, 62, a retired Army veteran and registered Democrat, had come to the forum with his wife, Charlene, to see if they could get help with some home repairs. But when he saw Ms. Murkowski, he stopped her to express his gratitude for her vote for Judge Jackson.“She has so much courage,” Mr. Townsend said of the senator. “She votes from her heart.”Later, at a pizza party at a local bar to benefit her campaign, Ms. Murkowski talked to supporters about her friendship with Mr. Manchin and long-gone titans of the Senate in both parties, name-dropping former Senator Daniel Inouye of Hawaii and quoting former Senator Ted Stevens of Alaska as she recalled a bygone era when camaraderie and common purpose tempered partisanship on Capitol Hill.Perhaps her own candidacy could prove there was still hope for that kind of politics.“You’ve got to demonstrate that there are other possibilities, that there is a different reality — and maybe it won’t work,” Ms. Murkowski said in the interview. “Maybe I am just completely politically naïve, and this ship has sailed. But I won’t know unless we — unless I — stay out there and give Alaskans the opportunity to weigh in.”Kitty Bennett More