Pierre stars in Maurice Sendak’s tale of a bratty boy who shouts “I don’t care!” at every turn—even when a lion is about to eat him. I applied the Pierre label to the rightists who sent the GOP lawmakers barreling into a government shutdown last fall—leaving the party’s already troubled reputation in tatters. That same Pierre-wing obstructionism now stands in the way of immigration reform, threatening Republican prospects for recapturing the White House in 2016 and beyond.

The nation’s fast-changing demographics pose a steep hurdle to a party that has lost the popular vote in five of the past six presidential elections. Among Hispanics—whose share of the vote is growing—the Democratic advantage grew from 22 points in 2006 to 48 points in 2012. A quarter of Hispanics are rock-solid GOP voters, and half are solidly in the Democratic camp.

That leaves a quarter up for grabs—but not if the GOP blocks immigration reform, which has become a high priority even among Hispanics otherwise drawn to a free-market message of opportunity, low taxes, and smaller government. The flip side is also true: Fully three-quarters of Hispanic voters would be more open to Republican candidates if they supported a legal pathway for undocumented workers living here, according to a poll released this summer by 10 prominent GOP pollsters under the auspices of the Silicon Valley advocacy group Fwd.us.

The Pierre wing’s border-security-only position is a loser on the national stage. Just listen to Grover Norquist, the GOP activist credited with making tax cuts the sine qua non of Republican messaging: “These dice are fixed, guys. The pro-immigrant, pro-comprehensive [reform] position keeps winning on this.”

The Fwd.us poll results were especially striking because they showed how out of touch the Pierre wing is with Republican voters. Those surveyed were asked to respond to a plan that included increased border patrols, employer verification systems, a guest worker program, and a pathway to legal status.

Republicans overwhelmingly supported the plan—including (get this!) 75% of those who fall under the Tea Party–conservative-white-evangelical umbrella. The key was avoiding the loaded word “amnesty.” Previous polling has shown similarly high support for a legal pathway, especially if there are conditions attached, like punitive fees or paying back taxes. A Pew Research Center poll last year found that 69% of GOP voters agree with the statement that undocumented immigrants are “hard workers who should have an opportunity to stay” in the U.S.

So why do so many GOP lawmakers act as if immigration reform is a political hot potato? Because the Pierre wing—and its boisterous radio talk network—is adept at spooking pro-reform Republicans. Florida senator and 2016 presidential prospect Marco Rubio practically went into hiding on the issue after touching off a vicious backlash from the right’s commentators.

Last year only 14 Republicans, including Rubio, voted for a comprehensive Senate immigration bill—but it was enough to pass that chamber and give business leaders hope that even with a more obstinate House, Congress would adopt comprehensive reform. Now, though, the White House’s hapless handling of 57,000 Central American children pouring into the country enables the Pierre wing to argue that the border is out of control. And the President’s executive orders on this and Obamacare give GOP lawmakers another excuse: “We won’t pass reform because we can’t trust the President to enforce the law.”

Long term, all that tough talk is political suicide. Republican Party leaders had the right idea after its resounding 2012 defeat. The party needs to broaden its tent, not batten down the flaps. And nothing since then has changed.

This story is from the September 1, 2014 issue of Fortune.