The Reagan Cabinet system spent a month wrestling with Japanese auto imports only to reach a non-solution so worrisome that at the March 19 Cabinet meeting Deputy Secretary of State William P. Clark had to come to the rescue.

Clark expressed the absent Secretary Alexander Haig's wishes that the State Department take the lead in trade talks with Japan, putting Special Trade Representative William Brock in a supporting role. Quick agreement meant that the State Department, not deeply involved until now, was coming in to execute policy after failure of Cabinet government to shape it clearly in its first serious test.

The administration consensus reached was U.S. support for a "unilateral voluntary quota" on Japanese auto exports, but nobody in the administration is quite sure what that means. Who will determine whether the quota -- "unilaterally" and "voluntarily" set by Tokyo -- is too high? The arrival on stage by the State Department ensures the Japanese will not be roughed up to "unilaterally volunteer" a quota smaller than they want.

That scarcely amounts to coherent administration policy. The new Cabinet system has failed to approach agreement on an issue where President Reagan himself is ambivalent. This suggests the chaos that would have ensued in the Cabinet over tax policy had not Reagan's position been so hard in pressing for rate reduction.

Instead of presenting the president with a unified recommendation on auto imports, the Cabinet exposed its own inherent splits. These disagreements transcend the auto import crisis and go to the basic question of governmental intervention in the economy.

Three millionaire businessmen in the Cabinet -- Transportation Secretary Drew Lewis, Commerce Secretary Malcolm Baldridge and Labor Secretary Ray Donovan -- favor intimate cooperation between government, business and organized labor in rebuilding the economy. Heads of those departments are not ordinarily involved intimately in drafting high economic policy, traditionally a Treasury function.

But under the Reagan Cabinet system, a task force on the auto industry headed by Lewis very nearly established a limit on Japanese imports as administration policy. The traditional economic policy-making departments, headed by the Treasury, belatedly counter-attacked against what they called Lewis' "runaway" task force.

By March 11, Lewis had been stopped cold; the Cabinet was hopelessly split. Treasury and the Office of Management and Budget wanted the issue taken to the president's desk in disagreement. But Lewis protested, contending this violated Cabinet government. Treasury Secretary Donald T. Regan was talked into it, giving the protectionists more time to finagle a low quota or other import limitation.

The complicating factor was that nobody was entirely certain what the president would do if the matter reached him in disagreement. "The old man is not all that tough on this one," one anti-quota White House aide confided to us. Worried about the sick auto industry, Reagan is torn between free trade philosophy and his sympathy for the U.S. companies and workers.

That led to a March 17 Cabinet-level luncheon out of which leaked inaccurate reports about capitulation by Treasury and OMB to a Japanese quota. The only agreement was on the "unilateral voluntary" quota now so deeply steeped in ambiguity.

Behind this is a genuine lack of consensus in the Cabinet over how economic policy should be conducted. Pro-quota partisans believe that willingness by United Auto Workers President Douglas Fraser to scale down workers' benefits in return for the Chrysler benefit should not go unrewarded. "We can't let Doug down," Labor Secretary Donovan said in one meeting.

"I think we ought to let Doug simmer in his own juices," an anti-quota Cabinet colleague said later. That fits the view of OMB Director David Stockman, who as a Michigan Republican politician remembers no favors toward him or Reagan from the UAW. But for their part, Lewis and other Cabinet businessmen regard Stockman as a non-businessman who does not understand the problems of troubled industries.

A smaller group has been trying to hammer out policy in line with the March 19 Cabinet decision: Haig, presidential aides Erwin Meese and James Baker and trade representative Brock.

But deep-seated ideological and stylistic differences between Dave Stockman and Drew Lewis, and between Treasury supplysiders and Transportation interventionists leave nobody happy. The dismay with which all parties look to the outcome suggests the limits of Cabinet government when the president himself is of two minds.