The Senate, narrowly rejecting a proposal to kill the Clinch River breeder reactor and brushing aside Democratic anti-recession initiatives, broke through a logjam last night and approved stopgap funding to keep the government from running out of money at midnight tonight.

The spending measure was approved, 72 to 26, as Senate Republican leadership sources said the White House, barring unexpected hitches in a House-Senate conference on the bill, appears to have no major problems with its provisions.

In the House, a proposed balanced-budget constitutional amendment gained ground as its backers won the 218 signatures needed to pluck the measure out of a hostile Judiciary Committee and force it to a showdown on the House floor. Details, Page A10.

The Senate's breakthrough on the catchall spending measure, which was passed last week by the House, means the bill can go to conference today and, theoretically at least, be passed and sent to the president for signature in time to prevent a disruption of government activities tomorrow.

Congress could then quit to campaign for the Nov. 2 elections this weekend, dashing conservatives' hopes for House action on the balanced-budget amendment before the elections. But Republicans were keeping up pressure for swift action on the amendment, preparing to use Democratic reluctance to bring the issue to a vote as a campaign issue if there is no vote.

Until mid-afternoon it appeared that the Senate, bogged down in a sea of proposed amendments and facing hours of debate on them, might have to go over until next week to finish its work, raising the possibility of time for action on the balanced-budget amendment in the House.

But the Senate quickly got back on track, moving with unaccustomed swiftness through the stack of amendments, after Majority Leader Howard H. Baker Jr. (R-Tenn.) and Appropriations Chairman Mark O. Hatfield (R-Ore.) threatened a seven-day extension of the government's spending authority that would have forced Congress to come back next week to finish work on the full measure.

"We simply cannot do the country's business this way. . . . One way or another we've got to get out of this," said an exasperated Baker in threatening drastic action, including another week in Washington, to get the stopgap spending measure through Congress without a government disruption.

While Republican leadership aides said the temporary extension was not aimed at allowing time for consideration of the balanced-budget amendment, some of its House backers, including Rep. Phil Gramm (D-Tex.), said they hoped Congress would stay in session long enough to act on the amendment, which, under House rules, would come to the floor Oct. 11.

The scene in the Senate until the logjam was broken was one of senatorial prerogatives carried to near-chaotic excess, as senators ignored leadership pleas for restraint in decorating the bill with pre-election ornaments and kept pressing for votes on their pet projects, which ranged from declaring Oct. 10 a national day of peace to extending unemployment benefits.

At one point, more than 50 amendments were pending, although they fell away rapidly after Baker's threat of another week in session.

In action on major amendments, the Senate:

* Rejected, 49 to 48, a proposal from Sens. Gordon J. Humphrey (R-N.H.) and Dale Bumpers (D-Ark.) to scuttle the Clinch River breeder reactor in Tennessee, which was defended by Baker as a "national project of major importance" but attacked by Bumpers as a "technological turkey." It was the second year in a row that Baker, defending his home-state project in a skeptical Senate, had to go all-out to save it. He won last year by two votes.

* Trounced, 60 to 37, a bid from Sen. Edward M. Kennedy (Mass.) and other Democratic leaders to create a public service jobs program for about 200,000 unemployed workers on roads, bridges and other similar projects, estimated to cost about $1 billion.

* Refused, on votes of 51 to 47 and 50 to 48, to liberalize unemployment benefits in as many as 31 states by suspending restrictions on extended payments to jobless workers. Proponents contended the proposal would help 2 million unemployed workers at a cost of $430 million, while Senate Finance Committee Chairman Robert J. Dole (R-Kan.) called it a "giant step backwards" that could wind up costing $3.1 billion in the coming fiscal year.

* Tabled, 50 to 46, a proposal from Sen. Ernest F. Hollings (D-S.C.) to ban procurement of MX missiles until the administration decides on a basing system for the new weapon, although research and development could continue. Rejection of Hollings' proposal came after Armed Services Committee Chairman John G. Tower (R-Tex.) denounced it in strong terms, contending there would be "dancing in the streets in Moscow" if it passed.

* Spurned Sen. Jesse Helms (R-N.C.), 62 to 37, when he proposed to prohibit unions from using money collected from compulsory dues for any political activities, a move that was opposed by many Republicans as well as Democrats who normally benefit from labor campaign financing.

Approved, 70 to 29, a Democratic-sponsored resolution opposing an idea being explored by the Reagan administration to impose a means test for Medicare benefits.

While Senate Republican aides said last night that the White House considers the overall spending measure to be within the budget, Budget Committee Chairman Pete V. Domenici (R-N.M.) told the Senate it would be $2.9 billion over budget if calculated on an annual basis.

However, the measure is only temporary, lasting until mid-December or until regular appropriations bills are enacted. So far Congress has passed only one of its 13 regular appropriations bills. At President Reagan's urging, Congress will return after the Nov. 2 elections to finish work on the individual money bills or, failing that, to pass another stopgap funding bill.

The only appropriations bill to win final congressional approval is the one for the Department of Housing and Urban Development and 17 independent agencies, including the Environmental Protection Agency and the Veterans Administration. A conferees' compromise was approved by both houses last night. White House officials have indicated that Reagan is expected to sign the measure.