The increasingly bitter conflict between the oil industry and the Carter administration reached a dramatic climax behind closed doors when Secretary of Energy James Schlesinger delivered what the oilmen consider an ultimatum.

With President Carter's energy program being shredded in the Senate, Schlesinger summoned the oil industry's top Washington representatives to tell them this: Support us on the crude-oil tax, or suffer severe consequences. The nation's oil barons have spent two weeks since then trying to figure out what to do.

While the administration contends that no ultimatum was issued, nobody denies pressure was applied. "It's hardball time in the energy program," one Schlesinger lieutenant told us. Thus, while Carter launches his "blitz" to mobilize public support, featuring his attack on "the biggest rip-off" during Thursday's press conference, his men are putting a strong arm directly on big oil.

That became clear when two top officials of the American Petroleum Institute - Frank Ikard, a former Democratic congressman from Texas, and Charles DiBona, a top energy official in the Nixon administration - were called to Schlesinger's office in the White House.

An exceedingly tense session followed. Schleesinger made clear that if the crude-oil equalization tax does not pass, the industry can count on lots more regulation and much lower prices -permanently.

Shaken, Ikard and DiBona returned to API headquarters and translated Schlesinger's comments into a threat to back the tax - or else. Top executive officers of the nation's big oil corporations immediately flew into Washington to debate new strategy.

Their inclination has been to seek a (compromise on the oil tax rather than fight it out with the administration. But the industry's friends in Congress want big oil to stand firm, feeling the President and his energy program are on the run.

Former Texas Gov. John B. Conally, the Democrat-turned-Republican now eyeing a possible 1980 presidential race, inadvertently infuriated host Gov. Meldrim Thompson and publisher William Loeb, the state's most powerful political influence, as a result of his inspirational talk to the Republican Governors' Conference in Bretton Woods, N.H.

Speaking to the governors Oct. 11, Connally marred an otherwise flowless performance by saying two things unacceptable to Thompson and Loeb: that he was a "free-trader" and that governors and gubernatorial candidates need not take a position on the Panama Canal treaties.

Thompson, a right-wing conservative who is a committed protectionist and implacable opponent of the canal treaties, was both angry and hurt. Both he and Loeb, though all-out Ronald Reagan backers in the 1976 New Hampshire primary, have been touting Connally as their possible 1980 presidential choice if Reagan does not run.

The evening before he spoke, Connally was specifically warned to watch his words. But perhaps overly stirred by repeated ovations he won from his Republican audience, he ignored the advice. The error will hurt him among conservative Republicans if he enters the 1980 Presidential primary.

Relations between the White House staff and the Cabinet took another downward step when top presidential aide Hamilton Jordan let it be known that - with two exceptions - he does not think much of President Carter's Cabinet.

Jordan's remarks came in a private White House session with top assistants of senators. Only two Cabinet members, he said, are worth much - Interior Secretary Cecil Andrus and Agriculture Secretary Bob Bergland.

If Andrus and Bergland are the best, one Senate aide asked, who are the worst? Jordan quickly replied that he could not single anybody out because one is about as bad as another.

The Senate aides were surprised that the President's right-hand man had priased two relatively minor Cabinet members while implicity undermining top policymakers in national security and economic affairs. The praise for Bergland, under heavy fire from farmers, was astonishing.

Word of Jordan's indiscretion quickly spread to the Cabinet departments, where there was instant outrage. "This is the kind of amateruish staff that has gotten the President in the trouble he is today," one assistant secretary told us.

Jordan's comments pinpointed the two Cabinet members closest to the White House staff - particularly Andrus. The former governor of Idaho coordinates everything with Jordan, which may explain his unexpected victories on tough environmental enforcement opposed by Secretary of Energy Schlesinger.