FORTUNE — As Washington backslides toward yet another man-made fiscal crisis, the group representing the nation’s top CEOs is sending a new and suddenly urgent plea to Members of Congress: Fund the government and raise the debt limit, now.  The appeal comes in the form of a letter from the Business Roundtable, the lobbying group representing leading chief executives.  “Failure to fund the basic business of government and adjust the debt limit in a fiscally responsible manner would risk both the immediate and long-term health of the U.S. economy and could permanently increase borrowing costs,” Boeing BA CEO Jim McNerney and BRT President John Engler wrote. “Even a brief government shutdown would have serious economic consequences and default, however temporary, would be calamitous.”  It’s an open question whether it’ll make any difference. The faction of Republicans who appear to be driving their party’s strategy going into the showdown have so far proven resistant, if not outright contemptuous, of appeals from the sorts of establishment authorities that traditionally have been able to help break through Congressional gridlock.  MORE: Bair: 5 reasons Republicans should support Janet Yellen  That’s both a symptom and a cause of the problem: The top-down dynamic that once prevailed in the institution, allowing Congressional leaders to hatch deals to move the business of governing forward, has eroded. It’s been replaced by liminal power centers accountable to no one but an increasingly feverish base.  The new order is plainly evident in the current mess, as ultra-conservative Republicans have rallied around an impossibly high ransom — the defunding of the incumbent President’s signature domestic achievement, his health care overhaul — as their demand for not pitching the country into economic oblivion. Their elected leaders so far have done little but look on.  Against that backdrop, the BRT missive reads as cautious to a fault. After raising the alarm about the stakes, the group goes on to encourage “both sides to adopt a new spirit of cooperation to achieve long-term comprehensive resolutions that put our fiscal house in order and clear the way for greater economic growth.” A separate letter the group sent President Obama on Monday traced the same argument, urging the administration and Congress to start talking. The letters follow a visit Obama made last week to a meeting the group held in Washington, in which he exhorted the corporate titans to apply “your influence in whatever way you can” to end the standoff.  MORE: Next for Germany’s Merkel: Fix Europe  The BRT’s messages join a growing pile of warnings from a corporate sector vexed by Republicans it supported with 72% of its contributions in the last election. A letter from the U.S. Chamber of Commerce to House members last week asked them to back off the threats. The National Association of Manufacturers seconded it. On Monday, former Sen. Judd Gregg (R-N.H.), now leading the Securities Industry and Financial Markets Association, a Wall Street lobby group, added a finer point. Writing in The Hill on Monday, Gregg decried that “the self-promotional babble of a few has become the mainstream of Republican political thought. It has marginalized the influence of the party to an appalling degree.”  So then again comes the question: What can business groups really do? In part, they can meet the new power centers where they live, literally. Letters are one thing. But ads in lawmakers’ home districts explaining to their constituents the economic consequences of a debt default would at least help throw the debate into some relief where it counts. Some Members of Congress may no longer feel accountable to their elected leadership in Washington. They’re all still responsible to those who sent them there. That’s leverage.
The corporate sector overwhelmingly backed Republicans in the last election, yet they can’t seem to get them to back down on their debt limit demands.