PAUL LAXALT of Nevada is one of the most agreeable people in the Senate and certainly its most amiable conservative. But his first order of business as the president's handpicked chairman of the Republican National Committee is a punitive action against a brother.

Sen. Bob Packwood of Oregon, who wants very much to stay on as chairman of the Republican Senate Campaign Committee, is Laxalt's target. Packwood riled the president last spring when he divulged that the president, when asked about deficits, skittered off into anecdotes about people who buy vodka with food stamps.

He also confided to the Associated Press that Reagan's plan for a Republican majority was "to build a party on white Anglo-Saxon males over 40."

It was "such potshots" rather than Packwood's noisy dissent on the AWACs sale that has made him a marked man. The president's antagonism has already been amply expressed: The White House erased Packwood's picture from a campaign brochure and yanked a Packwood fund-raising letter.

Silver-haired, gentlemanly Laxalt makes no bones about it -- Packwood has got to go.

"It's basic," he says, "When one takes a leadership position in either house, that leadership position requires working with the president. Once a policy has been established, or the leadership has got to go along."

The president's best friend in the Senate will nominate Sen. Richard Lugar of Indiana, a more conservative and discreet member, at the GOP organizing meeting next January.

Laxalt denies that Ronald Reagan put him up to it.

He simply "heard rumbles" last summer that Lugar was lusting for party office, and urged him to write a letter asking his fellow senators to "keep their powder dry" in the fight he was about to start. Some senators observed that Lugar, who was campaigning for reelection, had a nerve to bid for preferment before the voters had spoken. He was one of the few Republican incumbents who won handily in an election which many thought had vindicated packwood's warning about exclusionary politics.

To the general public, these intramural contests have all the significance and suspense of a playground scuffle or a power fight in the PTA. But senators regard these things with enormous gravity, and, in the course of deciding, reveal much of how they feel about each other, their prerogatives and the institution.

The outcome will, in this case, tell something about the president's clout on Capitol Hill. He hopes, although officially removed from the fray, to teach a salutary lesson to loose-lipped Republicans. A Lugar loss, on the other hand, would, at best, be interpreted as the Senate telling him to mind his own business -- and at worst, a signal that the Republican flight from him, begun in the campaign, willontinue.

Laxalt is not worried that the scrap will be divisive.

"It won't have a lasting effect. You know how civilized this place is. Remember the Heinz-Hatch fight for the same job? It was all forgotten in an hour."

Lugar is of the same mind. His good friend Packwood, has, he says, "a wonderful sense of humor."

Neither of the assessments take fully into account Packwood's bulldog profile and his bulldog spirit. He will not go gently.

He set to work last September and has, he claims, lined up between 26 and 29 votes. William Cohen of Maine got a call from Packwood, from San Francisco. Packwood, en route to China, wanted to know if Cohen's commitment, given weeks ago, was still good. Cohen assured him it was.

Lugar, who came back to town this week to fire up his campaign against his brother Republican, had indifferent luck with another member of the New England delegation, Warren Rudman of New Hampshire, who is officially undecided. Rudman likes both the rivals, and he doesn't mind changing campaign chairman, even though he thinks Packwood did "an extraordinary job" in the election. For Rudman, the issue is "outside interference."

"It is complicated by the fact that the White House is angry. I don't want any hate fests."

Lugar says that the contest should not be seen in Reaganesque terms. He says Packwood's vote count is exaggerated, by half. Lugar is counting on right-wing senators, six of whom are up in 1984 and don't want their dole coming from a pro-abortion chairman. John Warner, Roger Jepson and kingpin conservative Jesse Helms have already signed up.

It's a nice, small manageable fight -- a total universe of 54 votes -- and it has many fans, not just Ronald Reagan and his friend Paul Laxalt and the Republican senators.