In a paradox of ominous import for the Republican Party, Rep. William Cohen's bid for the Senate seat now held by liberal Democratic Sen. William Hathaway is threatened by a "constitutionalist conservative" candidate running as an Independent and backed by the national "New Right" political apparatus.

The objective: to transfer enough votes from Cohen to 30-year-old Hayes Gahagan, the Bible-quoting Independent, to guarantee the reelection of Hathaway, a liberal Democrat, over an impure Republican. That would wreck the Republican Party's most glittering prospect for defeating an incumbent Democratic senator and cripple Maine's GOP.

The paradox lies in the fact that moderate-liberal Cohen is supported by the Republican conservative establishment, including Ronald Reagan, New York's Rep. Jack Kemp and Utah's Sen. Orrin Hatch.

Praising Cohen's "excellent record" as a three-term congressman, Kemp on Aug. 14 wrote Maine businessmen urging "strong support" for Cohen. Reagan personally endorsed Cohen here Feb. 9.

Reagan went further. He telephoned Cyril Joly, a conservative who ran Reagan's 1976 presidential campaign here and resigned as Republican national committeeman to work for Gahagan, in a vain effort to dissuade Joly from breaking with his party and Cohen. Joly and State Rep. Walter Hichens are the only regular Republicans backing Gahagan.

At 38, Cohen is one of the bright young stars in the beleaguered Republican Party. But he is far from toeing the line of ideological purity demanded by the New Right, which wants to bar him from the Senate in the effort to mold the party to orthodoxy. "Case is gone and Javits is going. Cohen is the heir-apparent," one New Right leader told us privately, referring to long-time liberal Republicans Sens. Clifford Case and Jacob K. Javits. Case was defeated by conservative Jeff Bell in the New Jersey senatorial primary; Javits is 74 years old and may not run again for the New York seat.

Thus, when Gahagan formally announced his candidacy July 4, flanked by a replica of the Liberty Bell, the New Right offered immediate political goodies: Campaign indoctrination courses by the National Conservative Political Action Committee for both Gahagan and his campaign manager; a $1,000 contribution from NCPAC plus political "services"; $1,000 from the Committee for the Survival of a Free Congress; $500 from the Coors Employees Political Action Committe, campaign school for two Gahagan youth volunteers, courtesy of the Young Americans for Freedom.

Cohen now holds a margin easily comfortable enough to elect him - in a two-man race. But Gahagan may peel off 5 to 10 percent of Cohen's vote - possibly enough to reelect Hathaway.

That would have been laughable four years ago. That was before James B. Longley, running as a Conservative Independent, stunned this state by getting elected governor. Longley, who is not running for reelection, is neutral in the Senate contest, but his precedent-setting 1974 victory, coupled with his extraordinarily high popularity today, has set an enticing example. Maine's notoriously unstructed voters are disillusioned with politics-as-usual; Gahagan's simplistic pledge for drastic reduction of federal spending and interference in state affairs and against what he calls "the flim-flam" of traditional politics could have some impact on independent-minded Downeasters.

Cohen has formidable assets. "Billy is a smart politician," a top Hathaway operative told us, "but more important, he is a captivating courtier. Women look into his blue eyes, and he sweeps them away."

He has "swept away" maverick Democratic Mayor Lilian Caron of Lewiston, Maine's Democratic stronghold, who is campaigning for him there and in the heavily French American hamlets of the St. John River valley. William Rogers, former national commander of the American Legion, heads his veterans committe - a pointed rebuttal to New Right charges that Cohen is soft on defense.

Indeed, the New Right case against Cohen is spotty. He opposed the Panama Canal treaties (but not early enough to suit the New Right); the defense-oriented American Security Council rates him 78 percent (compared with Hathaway's 20 percent on "key" national security issues); he is an original sponsor of the Kemp-Roth tax reduction bill; he has serious doubts about the new strategic arms treaty.

The New Right's nightmare is not Cohen's record but the specter of a formidable moderate Republican in the Senate who is convinced that there must be room for diversity in the Republican Party - a conviction that elected Republican officeholders overwhelmingly share.