Backers of President Carter's bid for government reorganization authority began to tiptoe gingerly around the opposition of House Government Operations Committee Chairman Jack Brooks (D-Tex.) yesterday. They got through the first day of hearings on the bill without triggering an explosion from Brooks.

"I know we're going to be able to work this out," said Rep. Dante Fascell (Fla.), leading the majority of committee Democrats who are supporting Carter's bill over a rival proposal introduced by Brooks.

After Fascell leadoff testimony critics of the Carter proposal - Brooks among them - sharpened their attack on the aconstitutionality of the "one house veto" procedure it embodies.

But nothing that occured during the session challenged the assumption that the Carter bill - co-sponsored by 17 Democrats and four Republicans on the 43-member committee - would avoid the roadblock threatened by Brooks' opposition.

The Texas Democrat told reporters he planned one more day of hearings next week in his Legislation subcommittee and then would proceed to mark up the bill.

Carter's proposal, seeking resumption of authority that lapsed in 1973 during the dispute between Congress and President Nixon, would allow him to submit plans for reorganizing subcabinet agencies. Each plan would take effect unless vetoed by House or Senate within 60 days.

Brooks' alternative would leave the initiative in the hands of the President, but require an affirmative vote by both houses of Congress to take effect. It would prevent delaying a reorganization plan in committee for more than 45 days and would premit any member to call it up for a floor vote after that.

In an opening statement, Brooks said the Carter plan raises "serious constitutional questions . . . involving the separation of powers between the legislative and executive branches. It also raises serious practical questions about how much of its autority Congress should delegate to a President."

Bert Lance, director of the Office of Management and Budget, testified that the President was only seeking restoration of power granted to most Presidents since Harry Truman's time. He brought a legal brief from Attorney General Griffin B. Bell supporting te constitutionality of the one-house veto in reorganization matters.

That brief satisfied the Senate Government Operations Committtee, which last week unanimously cleared a slightly modified version of the Carter bill for Senate floor action.

But three witnesses told the House subcommittee that Congress was shirking its duty by avoiding the constitutional question.

Rep. Mendel Davis (D-S.C.) said that Brooks' plan was "the way our founding fathers meant a bill to be passed." And Antonin Scalia, a assistant attorney general in the Ford administration, also asserted the unconstitutionality of the Carter procedure, drawing a parallel to the method by which Congress let a pay raise for itself become law without taking a record vote.

"The pay raise was," he said, "the subject of widely critical comment - not for the substance of what it did, but for the process by which it was done . . ."

"It will indeed be disappointing," Scalia said, "if the first two major legislative steps taken by this new, post-Watergate Congress turn out to be the 'look-ma-no-hands' pay raise, quickly followed by the establishment of provision of 'blindman's-buff' reorganization."

Despite these arguments, the subcommittee Republicans, led by ranking Rep. Frank Horton (N.Y.), supported the Carter proposal. However, they expressed concern about a provision that would let Carter submit an unlimited number of plans to Congress, without restricting their subject matter.