Texas millionaire oilman and ex-Olympic yachtsman Robert A. Mosbacher, chief political fund-raiser for Vice President George Bush, was so angry that Howard Baker had arranged a White House ceremony for Sen. Robert Dole to endorse the INF Treaty that he took direct action.

Mosbacher telephoned members of Baker's former Washington law firm pleading for intervention with the chief of staff. The vice president himself took personal issue with Baker, hectoring him in a head-to-head conversation one source described as ''torrid.''

Baker and longtime aide Tom Griscom, the White House communications director, were infuriated. Making an enemy out of Howard Baker is no serious roadblock on the road to the nomination. What hurt Bush was what the incident showed about his own political style.

Dole operatives snickered over the anxiety exhibited by the Republican presidential front-runner. Not only did Bush's overreaction expand Dole's expected treaty endorsement into top political news, but the vice president also propelled himself into political infighting.

For all their cultural and cosmetic differences, Bush and Dole are strikingly similar in preoccupation with tactical politics and tendency to intervene in campaign management. Their duel over the intermediate-range missile treaty is geared more to presidential nomination politics than national security.

Bush strategists were euphoric when Dole delayed endorsement of the INF Treaty. They rushed out a ''peace'' commercial for Iowa television, climaxed by a freeze frame of the vice president shaking hands with Mikhail Gorbachev. Dole's Iowa forces pleaded with him to endorse too, and so did his campaign manager, William Brock. Unable to maintain a principled neutrality as conservative advisers urged, Dole looked like a trimmer as he drifted toward qualified endorsement. He ended up saved by his opponent.

As his predecessor as Senate Republican leader, Baker appreciated the importance of getting Dole behind the treaty and had been working on the press-room ceremony for weeks. Bush and Co., investing so much time in the summit, could not bear the thought of a tardy Dole sharing the spotlight with Reagan. Vice presidential aides blamed Deputy Chief of Staff Kenneth M. Duberstein, an avowed Bush backer before returning to the White House last March, for playing a double game with both sides.

The bitter infighting heated up with coincidental release of previously classified notes linking Bush to the Iranian arms sales. His operatives claim Baker promised the notes would not be declassified but reneged under pressure from Sen. Warren Rudman. Vice chairman of the Iran-contra investigating committee, Rudman switched to Dole after Baker, his first choice for president, said he would not be a candidate.

In fact, vice presidential chief of staff Craig Fuller rejects bad faith by Baker on that matter. Democratic counsel Arthur Liman and the committee staff, not just Rudman, insisted the document be declassified. Still, Baker was targeted by Bushites as furthering Dole's campaign.

What followed was Bush's dressing-down of Baker and what could be read as a threat in Mosbacher's calls to the law firm to which Baker may return. Gossip flew that Baker was trying to ingratiate himself so as to become Dole's running mate (a most improbable ticket) and was afraid a Bush presidency would install not Howard but another Baker, James, as secretary of state.

Bush's sentiments toward Baker are reciprocated by White House staff feelings about Bush and Lee Atwater, his 36-year-old campaign manager. ''We all know that adult supervision is needed in the Bush campaign,'' a senior White House aide told us. ''At the same time, the vice president cannot escape responsibility.''

If there are long-lasting ill effects for Bush from the pre-Christmas tiff, they will not stem from a Dole coup by virtue of his press-room appearance with the president or from coincidental news of Bush's renewed connection to the Iran-contra tar baby.

The problem for the vice president is that a small incident easily handled by praising Dole for backing the INF Treaty instead led to overreaction and intervention, political games-playing ill-suited to the real competition about to start.