President Reagan's quest for budget and tax cuts will get most of the headlines, but his mettle may be equally tested by the third front he has opened in his anti-inflation war -- his effort to cut the fat (and perhaps some of the muscle) from the federal regulatory establishment.

He grabbed the lead on the issue from Congress, when he announced on Jan. 29 what a White House press release termed a "60-Day Freeze on Regulations." In his economic message, he unveiled a new executive order that purports to be a "comprehensive program of regulatory management" to "reduce the excess burden of regulation." And recently he targeted 63 specific rules he wants trimmed or dropped.

White House press releases promise that Reagan will enforce his commitments, if need be, by intervening personally to pare back wasteful rules. If so, that will be a break with precedent. Even Republican presidents have usually stayed above the fierce battles of the past 15 years over health and safety regulation. Though dutifully hostile to Big Government in their speeches, Presidents Nixon and Ford turned the other cheek and signed the major regulatory laws that now bedevil American business, like the Clean Air Act, the Clean Water Act and the Occupational Safety and Health Act.

While the new White House says that Reagan will be a sterner boss, the question of his personal role is fudged by the fine print of the new regulatory management executive order and an accompanying Justice Department legal opinion. These indicate that the president will "guide and limit" his agency heads, but not necessarily "whoolly displace" their decision-making authority.

Whatever the details, Vice President Bush and OMB Director Stockman, who share responsibility for the issue, will face two broad questions. The substantive question: Which rules shall live and which shall die? And the turf question: Who shall be in charge? The White House, the agencies, Congress or the courts?

If the White House stumbles in trying to resolve these questions, the fallout could not only hurt the president politically but also weaken the institution of the presidency. Every important federal rule is a multiparty treaty resolving bitter, high-stakes battles among interest groups, agencies, courts and the Hill. Once it is widely perceived that the terms of these treaties may be reopened at the White House, Bush and Stockman and their staffs will receive much unsolicited help.

This process could get out of hand. If the press reports that lobbyists are swarming about the Oval Office to get their industry's most hated regulation on the president's hit list, Congress and especially the courts are likely to react. A federal judge could throw out an administration decision to cancel or cut back a rule. Then the whole procedure for White House oversight, and, indeed, the president's authority over the executive branch generally, could be undercut.

On the Hill, authorizing committees will become distressed if they think that "their" agencies are being effectively run by White House officials they did not confirm, do not know and cannot summon for an oversight hearing. They may retaliate by legislating the White House out of their particular piece of regulatory action. That is just what the Senate Commerce Committee did two years ago, when it gutted the president's authority to reverse Civil Aeronautics Board international airline route awards, based on suspicions that politics had affected White House decisions in some lucrative transatlantic and transpacific route cases.

Comparable institutional threats to the presidency are posed by regulatory reform bills now circulating on the Hill. They provide for legislative vetoes and for expansions of judicial authority that all administrations, Republican and Democrat alike, have opposed. Will the president break ranks with his Hill supporters (and renege on the Republican platform and his own campaign stands) by standing up for the prerogatives of his office? He has made it look like he wants to take charge. Does he mean it?

He could credibly do so, by pointing out that he can more easily cut out wasteful rules of opponents in Congress, the courts and the bar have no new procedural gimmicks to hang him up. But if he takes this course, some of his followers, watching Ronald Reagan side with regulators against their Capitol Hill tormentors, will wonder whether they are part of the New Right sequel to "Animal Farm."

In three years, all the rules that remain on the books will be Reagan's rules -- his political responsibility. Somewhere he will have to draw the line between fat and muscle, between useless red tape and needed public protection. Wherever that is, critics of regulation will have a hard time arguing that it is too far to the left.

So the probable, ironic result will be to legitimate the regulatory structure that remains when the current reductionist fevor is spent. Reagan's challenge is to take constructive advantage of this moment -- to be bold enough to make rational change possible, but sensitive enough to avoid shackles on his own ability to manage the government.