Ronald Reagan's lilting love song for Israel that drove Saudi Arabia and other pro-American Arabs states into a frenzy during the presidential campaign has become muted, a result of missionary work by insiders and Menachem Begin's miscalculation.

Prime Minister Begin's lack of gratitude for Reagan's ardent courtship of Israel reduced effusive pro-Israeli comments even before the election. That unwittingly bolstered efforts by advisers urging Reagan to take an even-handed approach in the Mideast.

These advisers seem to be winning out over Reagan operatives selling an all-out, pro-Israel position. If so, Reagan would have his only chance to gain mastery over the Arab-Israeli dispute and win what has eluded his predecessors: the West Bank settlement and Arab-Israeli peace that are inseparable from long-range American interests.

No change in the president-elect's attitude toward Israel has been proclaimed. In fact, some misleading evidence points the other way.

At Begin's gala Manhattan dinner last week, the head table bulged with top Reaganites: Richard V. Allen, national security adviser-to-be; William Casey, campaign chairman; former and perhaps future secretary of the Treasury William Simon; Republican National Committee Chairman William Brock. "I was asked by the Israeli ambassador," Allen told us, "and attended with pleasure. We are good friends."

But other Reagan insiders less sympathetic to Israel received no invitations. Gov. Bill Clements of Texas and George Shultz, a key transition adviser, are warning Reagan that he needs equally good relations with Saudi Arabia and Jordan. Reagan's Mideast policy experts are eyeing Jordan as the crucial link in Reagan's plan for peace on the West Bank.

Exponents of a balanced Mideast policy for the new Reagan administration are pushing hard for a shattering break with President Carter's Camp David process, which is clearly failing. Key elements in their plan:

Replace Camp David's design for West Bank autonomy with a territorial exchange between Israel and Jordan. That would abruptly shoot down Begin's claims to security rights over the entire West Bank.

Terminate Ambassador Sol Linowitz's mission impossible as special Egyptian-Israeli-U.S. go-between on the autonomy plan.

Assure Saudi Arabia that the new administration will quickly approve the bomb racks and fuel tanks the Saudis must have to give their F15 fighter-bombers worthwhile range in the Persian Gulf.

Adm. Thomas Moorer, a former chairman of the Joint Chiefs of Staff, warned Reagan just before the election that Saudi Arabia and other Persian Gulf oil states will never fully cooperate with the United States until the Palestinian homeland on the West Bank has been returned. That directly contradicts Begin's line that biblical authority gives Israel perpetual West Bank rights.

Begin's allies in the Reagan camp, who believe the fate of the United States in the Middle East will rise or fall with that of Israel, are battling back. This is their program in the Reagan transition: give Israel "veto power" over U.S. arms to the Arabs; push for a ruthless housecleaning of so-called anti-Israel "Arabists" from the Defense and State Departments; persuade Reagan to insist on U.S. military base rights in parts of the Sinai before returning to Egypt by Israel.

Until just before Election Day, the struggle for Reagan's Mideast policy was tilting toward the pro-Israel faction. It is now tilting the other way. Besided the efforts of the even-handed school, Reagan felt Begin's government failed to show proper gratitude for his effusive pro-Israeli positions early in the campaign.

In the three weeks before the election, while former Israeli defense minister Ezer Weizman went campaigning with Carter in Air Force One, Reagan scarcely mentioned Israel. One senior Reagan aide, conceding that Reagan had put a silencer on encomiums to Israel, claimed the reason was simply that "he said it all before."

In fact, it was during this period that Reagan heard alarm from advisers advocating even-handed treatment of Israel and the Arabs. Sen. Richard Stone, defeated for Democratic renomination in Florida and now being considered for a high State Department or diplomatic post, warned Reagan to beware Israel's claim to sovereignty over all Jerusalem. The private advice from Stone, who is a prominent Jewish leader: work at a Vaticanization of Jerusalem as a minimum favor to Saudi Arabia.

Stone's advice had an impact on Reagan. The battle to pull the president-elect from the brink of all-out support for Israel is far from over. pBut if it continues on its present path, there is reason for hope in the Mideast by the United States and its allies.