THE BUDGET FIASCO in the House of Representatives early Thursday morning was a severe embarrassment to the Democratic leadership. But for the administration, it's a warning that the rigid new budget process does not gracefully tolerate the kind of uncertainty and shock that President Carter is inflicting on it.

The budget process is still very new. While most of it went into effect last year, the full apparatus is only now taking hold. Until two years ago Congress handled appropriations piecemeal, leaving the structure and strategy of the budget pretty much to the President. Under the new process, Congress has to vote on all the key totals - revenues, spending and deficit. It's a severe discipline but, if the 535 members of Congress can keep it working, it will give them the last word on the nation's fiscal policy.

The budget resolution got off to a bad start Tuesday, with a good many of the House Democrats still sore at President Carter for having abandoned the $50 tax rebate. Then the administration mounted an intense lobbying campaign to restore the $2.3 billion that the Budget Committee had cut from defense spending. Mr. Carter himself asked for the money, and finally he got it. But perhaps the White House didn't fully understand that the $2.3 billion cut in defense was part of an intricate cat's-Cradle of compromises and deals touching nearly every other part of the budget. The people who lost on the defense cut immediately cried treason. When the final vote came on the budget resolution as a whole, there were only 84 votes for it - and 320 against.

The Republicans voted against it, almost to the last persons, because they thought the deficit too large. At the same time Democrats deserted in droves on grounds that - after the withdrawal of the rebate and the restoration of the defense outlays - there had been too many concessions to business and the conservatives. Now the Budget Committee has to go back and patch together another set of compromises.

So far there's been no irremediable harm done. But these dramatics suggest that it may be harder to pass budget resolutions now that the President is no longer a Republican. The House Republicans always hesitated to oppose Mr. Ford flatly, but they will have no compunctions about exploiting the deficit as a weapon against Mr. Carter. He is beginning to lose support among those congressional Democrats who consider him to have moved much too far to the right on money matters. But he can't expect much support among the Republicans, despite his prudence. Mr. Carter got his way on the rebate and on defense spending - and, as a result, there's no majority in the House for his budget.