The two Republican candidates for mayor of the District of Columbia have their own lighthearted ways of venting the frustrations of running for office in an overwhelmingly Democratic town where no one--including some key Republicans--seems to notice the GOP primary.

Candidate E. Brooke Lee Jr., a 64-year-old realtor with a politically prominent family name but questionable D.C. residency, quotes Richard Nixon.

"I'm white, I'm Republican and you probably don't know me because I haven't been here very long," Lee said at a forum at Howard University. "So you might ask, 'Why should I elect you mayor?' Here is what I say to you: 'Trust me.' "

James E. Champagne, a 39-year-old Kennedy Democrat-turned-conservative who is long on ideas for changing the city but short on campaign cash, tells of dreaming that he is touring Heaven with St. Peter and at one point is told to be quiet as the two approach a wall.

After they pass the wall, Champagne says, St. Peter explains: "He said, 'That's where the Catholics are, and they think they are the only ones up here.' Well, it's like that in the District of Columbia. For too long, the Democrats have thought they're the only ones who can be mayor."

These often are lonely days of wry humor on the Republican campaign trail.

When former D.C. school superintendent Vincent E. Reed earlier this year declined to seek the nomination for mayor, he snuffed out Republican hopes of upsetting the Democrats in November.

Since then, the local Republican Party organization has chosen to remain at arm's length from the contest, refusing to endorse any candidate before the Sept. 14 primary--and leaving the field open to a pair of little-known candidates.

"Frankly, I don't think the majority of people consider either of these candidates as viable," said one member of the D.C. Republican State Central Committee involved in candidate recruitment.

"Champagne is a bright young man, an attractive guy, and he may have a future," this source said. "Brooke seems to be running on his family name, and I don't know what the hell he's done."

Ward 3 Republicans in particular are wary of getting involved in a potentially divisive GOP race for mayor that might harm their bid to win the D.C. City Council seat now held by Democrat Polly Shackleton, who faces one of her toughest challenges ever in the Democratic Party's potentially divisive primary. About half of the city's 22,951 registered Republicans live in Ward 3, west of Rock Creek Park.

"The Ward 3 organization is dedicated to electing Lois De Vecchio Republican-endorsed candidate for City Council , and anyone who gets in the way of that will have trouble with the Ward 3 organization," said Bob Carter, chairman of the D.C. Republican Party.

Said John D. Garst Jr., chairman of the Ward 3 Republican organization: "My feeling about the mayor's race is to just stay on the fence."

Lee and Champagne gamely maintain that their candidacies could catch fire and catapault them into the mayor's office this fall, despite the Democrats' 9-to-1 advantage in registered voters.

"If Marion Barry bears down on Democratic voters as hard as he is with the help of 30,000 city employes, some of them are going to be so mad they'll come to us," Lee said. "This is a great time for a Republican to step up to the plate and hit a home run."

Champagne is less optimistic about his chances, but says that his candidacy is important in promoting a two-party system here. He blames the Republican State Central Committee for many of the problems facing the GOP mayoral candidates.

"The bottom line is that people think I'm crazy for trying to do this," Champagne said. "Some knowledgeable people have told me: 'You're not only running against the Democrats, you're running against your own party.' "

Joseph N. Grano, a 36-year-old lawyer and unsuccessful City Council candidate, also has filed as a candidate in the Republican primary. But his campaign is linked chiefly to his efforts to save the 183-year-old Rhodes Tavern and he does not intend to spend more than $250 campaigning.

The history of D.C. Republican mayoral politics has been a dismal tale. Apart from its obvious numerical disadvantage, made worse by the Watergate scandal, the local Republican Party organization here has suffered from indifferent and unimaginative leadership. And good candidates have been hard to find.

In 1974, the Republican Party did not actively oppose incumbent mayor Walter E. Washington, a Democrat, who won easily over the Republican nominee, Jackson Champion, 79,605 to 3,501.

Four years later, the GOP enthusiastically endorsed Arthur A. Fletcher for mayor. Fletcher, a former Nixon and Ford administration official, was expected to mount a strong challenge to the Democrats and raise considerable funds. Instead, his campaign fizzled and he was clobbered by Democrat Marion Barry. Fletcher received less than 30 percent of the votes cast.

This year, the Republicans tried to woo Reed, the highly popular former school superintendent and assistant secretary of education in the Reagan administration, with promises of substantial financial backing.

But Reed concluded that he was not a politician and declined the invitation. He subsequently went to work for The Washington Post as a vice president for communications.

Fletcher, who runs a management consulting firm here, said this week that Reed was the only potential GOP candidate with broad enough appeal to attract Democratic votes and raise substantial campaign funds.

But Fletcher noted that, even under ideal conditions, electing a Republican mayor in Washington "is like trying to climb Mt. Everest in tennis shoes."

Michael Gill, a Republican national committee member and chairman of the local party's candidate search committee, said that he tried to persuade Lee and Champagne to team up, with one switching to the race for chairman of the D.C. Council. Both men flatly refused, according to Gill.

Champagne has worked as a teacher at the Sidwell Friends School here, a federal manpower and trade specialist and as a speech writer for the Chrysler Corp.

He says he is bent on elevating the mayor's race from a popularity contest to a serious debate on ways to reduce the city's spending and taxes while improving its services.

Champagne also says that he would improve the school system by eliminating tenure for teachers and forcing them to sign performance contracts. And he would try to impose a 24-month limit on the time that poor families could expect to receive welfare payments.

Both Champagne and Lee believe that the District government can be made more efficient by hiring private firms to manage many key services, including water billing and garbage collection.

"Government employment must be patterned more along private industry lines," Champagne explained. "The government shouldn't be the employer of last resort, which it has become."

Lee, a former executive of the Scott Paper Co. and the brother of Blair Lee III, the former acting governor of Maryland, boasts that he is part of an Ivy League "old boy" network that could help him bring new jobs to Washingon if he is elected.

Lee said he would call on old Harvard classmates who went on to head Fortune 500 businesses, and urge them to set up new operations here.

A former World War II airborne infantry commander with a straight-from-the-shoulder delivery, Lee gravely warns his audiences that "we may be in for a hot summer" of civil unrest because of widespread unemployment and that it is time to elect a Republican mayor who can offer hope to the jobless.

Lee also promises to crack down on street crime and illegal drug trafficking, and says that he is not afraid to follow the police into the streets to see that it is done properly.

"I didn't walk through the airborne infantry to be scared walking the streets," Lee told a gathering of 25 supporters last week in a Northwest Washington home.

Lee and his aides have lobbied the Republican State Central Committee, which is heavily loaded with Ward 3 residents, to end its neutrality and throw its support to him.

Lee claims that 41 of the committee's 77 members will vote to endorse him at a June 8 meeting--an endorsement that he says is essential to raising the funds necessary to mount a respectable campaign.

Lee's aides say that they have raised $25,000 to $30,000 so far, a drop in the bucket compared to the hundreds of thousands of dollars that have been raised by the major Democratic candidates. Champagne, by comparison, has raised only $6,000.

"You have to have one candidate of the Republican Party to raise money," Lee said. "A lot of people aren't going to contribute until there is one candidate."

But some party officials say they doubt that Lee has adequate support on the state central committee to win the endorsement and that he may be headed for an embarrassing setback.

Even if Lee succeeds in winning the endorsement and the primary contest, Champagne said that he intends to file suit to challenge Lee's residency qualifications. "At best he's a carpetbagger," said Champagne, a native of Northhampton, Mass., who has lived in Washington, on and off, since 1967.

Lee, who comes from a long line of Maryland politicians and civic activists, maintains a home with his wife in Falmouth, Maine, where he voted in the last presidential election.

He claimed his mother's house at 2340 Kalorama Road NW. as his D.C. residence when he registered to vote in the District last Dec. 18. Lee's name was not officially added to the city's voting rolls until Jan. 2.

D.C. election law requires that candidates for mayor must have "resided and been domiciled" in the city for at least one year preceding a general election or special election.

But the definition of residency is open to broad interpretation by the courts, according to a lawyer for the city elections board, and Lee and his aides say they are confident that he qualifies as a candidate.