As solemnly as a couple picking out an engagement ring, or a banker and borrower agreeing on collateral, Jimmy Carter's White House and Saudi Arabia's royal family have settled on a symbol for their deepening attachment. In the tradition of power relationships in the Persian Gulf, the symbol is the latest, most destructive fighter-bomber in the U.S. Air Force inventory.

The warplane is the F-15 all weather twin-engine interceptor, and the Saudis want 60 of them. Faced with congressional opposition, the White House is quietly shaping a campaign to get the sale approved as the centerpiece of its effort to assure the Saudis that behind all the pretty words, Americans really do care.

"Every signal we get is that the White House considers this sale more important than any other weapons transfer it envisions," says a member of the official U.S. foreign policy community. "They will go to the mat on this one, even if all hell breaks loose on the Hill, because they need it to cement the Saudi relationship."

That campaign is taking shape only six months after Carter promised to reduce the importance of arms sales as an instrument of American foreign policy. Yet, administration advocates of the deal argue pesuasively that the F-15 sale would not affect the military balance in the Persian Gulf or Middle East. They conclude that its importance lies in politics and national prestige for a nation that does not have a front line fighter.

Moveover, the active backing for the sale stands in sharps contrast to administration's firm opposition to the wiches of the shah of Iran to buy another new American product, the F-18 attack plane. Reports fro Iran suggest the shah is not amused by the contrast.

Behind the projected struggle over the F-15 for the Saudis lies a story of rapidly changing perceptions in Washington of Saudi power and usefulness in a changing world. The changes are occuring not only in the White House but also in the Pentagon and on Capitol Hill, where the Saudi case is getting increasing support.

Under Richard Nixon and Gerald Ford, Iran was clearly the most favored American ally in the Persian Gulf and had a blank check to buy the weapons system the shah wanted. Massive deliveries of F-4 Phantom warplanes, then the Air Force's hottest item, embodies Nixon's early and strong commitment to the shah.

Now, despite his past criticism of Henry A. Kissinger's realpolitik approach, Carter himself is using the same instrumentality - a weapons sale - to put a stamp on his administration's primary Persian Gulf relationship. Moreover, a series of reviews growing out of a National Security Council Policy memorandum on arms transfer is crystalizing Carter administration policy in the Gulf. They evidently point toward Saudi Arabia's increasingly displacing Iran as the American linchpin in the Gulf.

To a great extent this is due to the shah's own determination to be more independent of Washington. While Saudi Arabia is still laboring to complete 80 pe cent of a defense plan drawn up in 1974 not by Saudi planners but the Pentagon. Iran is pressing hard for items Washington is at best reluctant to send.

There is also the obvious, documentable importance of the Saudis' immense petroluem production, and the royal family's skillful playing to Carter's presidency by fighting against a price last December.

"The Saudis gave Carter the best possible Christmas present even before he became President, and I don't think he forgot it," says one professional anyalst of the relationship. "Neither did the shah, who made the same kid of announcement in Washington in November."

There are hints here of a large Carter administration view of coping with the world more through economic and financial power rather than through the kind of pure military might Iran represented for the Nixon Doctrine. In an emhryonic stage at best,a Carter Doctrine in which Saudi Arabia would play a dominant role appears to be taking form.

The first parts of this series have described what is in effect the building of a new sub-system of client states in northern and central Africa, the Middle East and Asia, with primary loyalties that run to Saudi Arabia. With one vital exception - the Arab bloc's opposition to Isreal - the interests ofthose states are part of the shared U.S. Saudi goal of limiting Russian and radical influence.

Added to the oil and petrodollar flows the Saudis have pledged to keep open to the United States this sub-system represents part of an enormous quid in the new strategic equation. The F-15is the symbol for many Saudis of the most important American quo , the implied security commitment to protect the country and the royal family from stronger, potentially greedy neighbors and foes in the region.

Administration officials pushing the sale accept that symbolism as well. They argue that denying a relatively small number of an advanced American fighter-bomber to the Saudis - who have been assured by Nixon, Ford and a plane - would do enormous strategic damage to the special relationship the United States and Saudi Arabia have fashioned.

The fight in Congress, which the administration decided to delay at least until the next session after taking a sampling of the mood this year, will be decide ultimately on that premise.

The debate is also turning into a test of American intentions in the Middle East conflict. Saudi officials are convinced that "the Zionist lobby" is the only important force working against the sale, and they would interpret a retreat by the administration or a defeat in Congress as proof that American policy on the Middle East cannot be moved in their direction.

Neither the growing Saudi help for U.S. objectives abroad not the dramatic movement toward peace initiated by Egypt's Anwar Sadat, the most important Saudi aid recipient, has erased the resentment and fears raised in Congress by the Saudi role in financingthe continuing re-equipment of Arab armies since the 1973 war.

Administration officials assert that two F-15 squadrons, designed to replace a rapidly disintegrating force of British lightnings, would give Saudi Arabia only a modest increase in defensive ability. The deep penetration of the small Saudi military establishment by American commercial and military technicians limit the potential use of the weapons against American wishes, these officials suggest.

The plane, in their view, offers Saudi Arabia more political and prestige benefits than military advantages. Smaller oil producing neighbors like Abu Dhabi and Oman have already put highly sophisticated French and British fighters into service, while the Saudis have stuck to the pentagon program of building up a force of limited-range U.S.-made F-5s.

Arguments on the other side are being marshalled for the fight. Staffers for Sen. Hurbert H. Humphrey (D-Minn.) have already triggered a General Accounting Office study that concludes that the Saudis may not be able to absorb the F-15. This week, Sen. Henry M. Jackson (D-Wash.) issued a study that endorsed continuing U.S. support for Iran's military buildup but withheld a similar endorsement for Saudi Arabia.

The appearance of such studies is in itself one sign of the new importance on Capitol Hill of the Saudis, who for the first time are making an effort to have their important visiting officials get together with members of Congress as well as excutive, branch officials.

In 1972, Jackson made a visit to the Saudi capital of Riyadh and was told he was the first Senator to visit in five years. In the past 24 months, however, congressional delegations have increasingly beaten a path to the desert city.

"In a historic context, Saudi Arabia seemed pretty much to be a massive sand dune until came into the picture, "Sen. Mark O. Hatfield (R-Ore) said in an interview. "Today there is an exploding growth of interest in the Saudis. We have less reliance on the shah than we had, although there is still the recognition that the shah has the only force in the Gulf."

Remarking that there is "still a reluctance to say publicy what many are saying in the cloakroom," Hatfield said that "low, moderate voice of Saudi Arabia" has helped convince senators that "we have a multiple interest in the Middle East. The Israelis have had a monopolistic view. Now, as in the case of the shah, there are shared interests. It is now all interrelated, and it is longer just one country dominating our interests."

"Israel's supporters in Congress are nervous about the clout the Saudis have now," said Rep. Lee H. Hamilton, (D-Ind), chairman of the Europe and the Middle East subcommittee of the House Committee on the International Relations. "There are some crunches coming. The pro-Isreal group in Congress will strongly object to the F-15 sale, as will people like myself who have reservations about the flow of arms in the area."

While they will not discuss it for quotation, U.S. officials with long experience in Saudi affairs report that there is a final silent joker in the F-15 pack.

"The fact is that the greatest constraint on a strong military force in Saudi Arabia is the royal family," said one analyst. "They have watched growing armies overthrow their royal Arab brethren, and they don't want to risk that temselves.

"They want some modern weapons and an army so they can hold their heads up in Arab world and be credible," the anayst continued, "They have U.S. military mission there largely to bolster that credibility. But they know that in the end it is the U.S. that would have to save them from an outside threat, and that is why symbols of commitment are as important to them as they are to people like the West Germans."