Extreme caution inside the Carter administration over Israel's new right-wing regime, headed by Menachem Begin, is clearly evidenced by the delay in inviting Shmuel Katz, Begin's intimate political adviser and friend, to the White House during his visit to the United States.

A campaign to break down that caution and enlist President Carter as a full-fledged participant in traditional American-Israeli friendship is now at flood tide. But the gap is too broad for semantic bridging. Thus, some of Israel's most realistic backers here are looking to outside events - specifically, a reconcillation of Arab states with theSoviet Union - to push political opinion here back in Israel's direction.

On a lengthly visit to the United States to stir support for Begin, Katz used as emissary to sound out what would happen if he asked for an appointment with Zbigniew Brzezinski, Carter's national security aide. Pointedly, the chance to issue an invitation was passed up.

Before returning here later this week, Katz decided on Tuesday to take the initiatiave and seek an invitation from Brzezinski. It was duly offered and the two are now scheduled to meet Friday. The one-week delay in the Brezinski-Katz talk is blamed on the fact that not until Monday was Begin formally asked to form a new government as prime miister. In point of fact, however, at any earlier period the mere hint of a White House cold shoulder for the emissary of a prospective Israeli prime minister would have been unthinkable.

This understandable coolness stems from Begin's blunt post-election edict that the West Bank belongs to Israel. So, Israel's key backers here are beating a track into the White House on this errand: Persuade Carter through his advisers and aides that Begin and his hawkish Likud coalition are really no different from outgoing Prime Minister Yitzhak Rabin and the Labor Party.

One leading pro-Israeli lobbyist has made no fewer than three visits to the White House (though none to see the President) since the Israeli election. His message is the one that Katz has been delivering here and in New York: Begin accepts United Nations resolution 242, which calls for the return of "territories" (unspecified) captured by Israel in the Six Day War, in exchange for real peace and for what the Israelis call "defendable" borders.

Begin is portrayed as also accepting Carter's insistence on the return of all but "insubstantial" portions of the West Bank. He does so, Begin's agents here emphasize, because the President invariably adds that final borders must be a matter between the parties - Israel and the Arabs.

These semantic efforts to bridge the very wide gap between the President and prime minister will fall short. Carter is the fourth President to take an ironclad position against any new Israeli settlements on the West Bank; Begin's first call after his election triumph was for "many" new Jewish settlements on the Arab West Bank.

So, political assets from events beyond the narrow Washington-Jerusalem channel are perceived by the pro-Israel bloc as essential to bridge the Carter-Begin gap, and the first may be the forthcoming meeting between Egyptian Foreign Minister Ismail Fahmy and Soviet Foreign Minister Andrei Gromyko.

Their meeting suggests relations between Egypt and the Soviet Union are in for a major overhaul, perhaps the beginning of the end of the fractured period starting in 1972 when President Anwar Sadat kicked some 14,000 Soviet "advisers" out of Egypt.

Sadat desperately needs Soviet spare parts and weapons to have any chance at all to match Israel's U.S.-equipped forces. With Congress limiting Egypt to transport planes, Moscow is the only alternative.

Talks in Vienna between Fahmy and Gromyko have been rumored for several weeks, but Fahmy may now have decided to go to Moscow. That locale would heightten the dramatic impact of the talks and the change in Israel; but it would also have predictable political impact on the U.S. Congress.

In truth, however, Sadat and other Arab leaders could not be expected to show the same exemplary restraint of the past three years over U.S.-mediated peace talks with an Israeli leader who proclaims West Bank settlement as a cardinal aim. Unless Carter can convince those Arab leaders that he can change Begin's mind - or, far less likely, that Congress will use its power to twist Israel's arm - other Arabs will do what Sadat is doing in sending Fahmy to Moscow: look to the Soviets for help.

That is the source of the other political assets for Israel. The closer the Arabs feel compelled to move toward Moscow, the greater will be Israel's political gain here. Accordingly, Begin's election may have started a new vicious circle, undercutting the Arabs' faith in the United States and sending them to Moscow. That in turn would intensify pro-Israel and anti-Arab emotion throughout the U.S. government, and especially in Congress.