DES MOINES -- No sooner had George Bush won his shouting match with Dan Rather than their duel was converted by his managers here into a lever to pull victory from defeat in Iowa and all but clinch the Republican presidential nomination.

Relentless Bush operative Rich Bond trotted out state campaign chairman George Wittgraf, an amiable Iowan who for two years has dutifully hurled hatchets handed him by national headquarters. Dozens of radio stations received recorded statements by Wittgraf comparing Rather with Sen. Robert Dole for twitting Bush about Iran-contra. Anybody listening to their car radios along Iowa's snowscape could hear the ineffable anchorman linked with their fellow midwesterner.

That is part of high-stakes hardball here that conceivably could wrap up the nomination early -- at the Feb. 8 caucuses. Dole still leads, but not by much. If Bush barely nips him, it will be hard to stop the vice president.

Were Iowa a primary instead of a caucus, there would be little doubt of Dole's victory. Even Bush insiders concede that the senator's ''one of us'' commercials reinforced his regional popularity, unaffected by revelations that he has become a millionaire. Among party activists, Bush suffers from identification with the likes of ex-Republican national chairman Mary Louise Smith and Rep. Jim Leach, who are stigmatized less as liberals than as elitists. At the same time, the numerous anti-Reagan Republicans here prefer Dole.

But the outcome of Bush vs. Dole depends on which side gets its people out on a cold winter night. Neither the candidate commands fanatical devotion or presents enticing visions. It is a struggle of organizations. Tim Synhorst, a political operative for Sen. Charles Grassley, has built an impressive but untested organization for Dole. In contrast, Bush's roots here go back to his 1980 eyelash upset of Ronald Reagan.

Bond, Bush's deputy national campaign manager and author of the 1980 surprise, was brought back here kicking and screaming last October after the vice president was surprised by Pat Robertson in the straw vote at Iowa State University. Since then, Bush's campaign here has borne the Bond trademarks: tight organization and political hardball.

So far it hasn't been enough. Trackings of caucus-attender sentiment show Bush briefly closed the gap during Reagan-Gorbachev summit week, then fell back. Heavy television advertising boosting Bush is beginning, to be followed by final-week sniping at Dole's personal finances.

Dan Rather is the new factor. Dole's nightly trackings again showed his lead narrowing. That includes support from conservative voters who rally to Bush as the president's defender on Iran-contra.

Rep. Jack Kemp has been gaining ground with those voters by preaching the Reagan Revolution in undiluted form, while Bush and Dole join Democrats in harping on deficit reduction. Neutral Gov. Terry Branstad echoes a widely shared opinion that Kemp might well pass Robertson in Iowa for a needed third place finish. Kemp's own operatives lament that his name recognition in the state still hovers at 50 percent.

He is making progress with senior citizens ("my opponents want to freeze or cut Social Security") and battling Robertson hand-to-hand for evangelicals. Robertson in response has opened an anti-Kemp media effort. Radio spots on Christian stations by Michigan state senator Ed Fredricks contend Kemp ''has just delivered Michigan to Bush'' and charge that it was always clear that he would ''sell us out.''

The Bush campaign also will harp on this weekend's Michigan convention. Bond plans to use the expected win to show worried party regulars that the vice president can handle Robertson's evangelical interlopers in a state where Dole was shut out for delegates (in fact, he did not compete there).

Bush was not scheduled to attend the Michigan convention. While Kemp and Robertson go to Grand Rapids and Dole campaigns in New Hampshire, the vice president will be slogging through Iowa. His managers hint at surprises in the last week of campaigning, but they really hope for nasty weather the night of Feb. 8, a climate in which only the well organized would prosper.