Mr. Katz - Schmuel Katz - came to Washington this week, all the way from Tel Aviv. And what he got was an immersion in American red tape that engaged the interest and prerogatives of the State Department, the Justice Department and Zbigniew Brzezinski.

In the end, the U.S. government concluded that Mr. Katz for 10 days would be a foreign agent. At home they call him a press agent for Menachem Begin, who is expected to become the prime minister of Israel.

Quickly:

Katz's mission is to do a little image-building for Begin, who is known here and at home as a hardline operator vis-a-vis the Arab world. Katz, since his arrival, has been knocking on a lot of doors, trying to assure politicians, diplomats and journalists that his client is a reasonable man.

When the Justice Department got wind of his arrival - some say the Arabs tattled - the Israelis were informed that Katz would have to register as a foreign agent.

"We have great doubts," said Avi Pazner of the Israeli embassy, "whether a temporary visitor has to register, and we told the State Department."

The State Department was inclined to agree, and the matter, reportedly, got bucked on to the President's national security adviser, Brzezinski.

Conversation followed between Brzezinski and Attorney General Griffin B. Bell and the State Department and the foreign agents' registration section of Justice, and finally it was resolved.

Late yesterday Katz registered as a foreign agent. He will lose that title when he returns home next week.