Despite the political hoopla and the lopsided public opinion polls, the national campaign for a constitutional amendment on the federal budget faces growing prospects of being derailed.

A backlash from the Washington political establishment and closer scrutiny of the campaign's reported success to date are raising serious doubt about what seemed, just a few weeks ago, a near-certainty: the convening of a constitutional convention to propose a balanced-budget amendment.

There's no doubt that the convention campaign, propelled by a charismatic national leader, California Gov. Edmund G. (Jerry) Brown Jr., and a broad grass-roots network, has acquired considerable momentum. Polls say that more than two-thirds of the American people support a balanced-budget amendment, and that most favor calling a convention, if necessary, to propose one.

The National Taxpayers' Union, the Washington-based lobby coordinating the drive, said Friday that 28 of the required 34 states have approved a resolution asking Congress to call a convention. The group said two more states, Indiana and Montana, are likely to do so within two weeks.

But the campaign's recent successes may be misleading. Political and legal realities suggest that chances are slight that a constitutional convention will be called. The drive probably will prompt some action from Congress on the budget issue -- but it is likely to fall far short of a constitutional requirement for a balanced budget.

Proponents of the convention drive face three obstacles.

They have an uphill fight to win the approval of the additional states needed to reach 34. If they do accumulate 34 state resolutions, Congress is likely to hold many of them invalid. And if proponents ever approach 34 valid resolutions, Congress is likely to adopt legislation or propose a mild amendment in order to avoid calling a convention.

The Constitution says amendments can be proposed by a two-thirds vote in Congress or by a convention called at the request of two-thirds (34) of the states. In either case, any proposed amendment would have to be approved by three-fourths (38) of the states to take effect.

The drive for a constitutional convention started slowly four years ago and then picked up steam last summer in the furor surrounding California's Proposition 13. Brown's declaration of support six weeks ago added new impetus.

But Brown's support, and the attention it won for the convention campaign, spawned a counterattack from Washington. House Speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.) established a task force on the issue. Last week the task force chairman, Rep. David Obey (D-Wis.) sent a tough letter to every governor warning that federal aid to state and local government is sure to be a victim of a balanced-budget austerity wave.

When the governors gather here for a conference this week, congressional leaders will be among them for anticonvention lobbying -- although this effort may be offset by the personal lobbying of Jerry Brown.

The National Conference of State Legislatures has scheduled a meeting here this week, too, and the legislators will be welcomed to Washington by Sen. Edward Muskie (D-Maine), chairman of the Senate Budget Committee and a caustic critic of the balanced-budget proposal. Muskie will reinforce the message that local aid will be slashed if the states force any budget-cutting action on Congress.

To counter the Obey-Muskie argument, some proponents of a constitutional amendment have proposed adding language that would prohibit any significant cuts in local aid. That idea, in turn, hardens the congressional leadership's opposition to any budget amendment.

Two Senate committees will hold hearings early in March on the wisdom of holding a convention and on the need for a constitutional regulaton of the budget. The witness lists suggest that the sessions will portray a constitutional convention as an invitation to disaster and a balancedbudget requirement as an economic straitjacket.

On the House side, Chairman Peter W. Rodino Jr. (D-N.J.) of the Judiciary Committee has planned a cautious, drawnout investigation of the various proposals for an amendment limiting government spending. "Rodino has launched a turtle," said one aide to the House leadership.The purpose is to create enough evidence of congressional activity to deter additional legislatures from demanding a constitutional convention.

President Carter and Vice President Mondale are as strongly opposed to a balanced-budget convention as the congressional leadership, but the White House has not yet been active in the opposition campaign. That may be starting to change.

Political advisers are urging the president to take a visible position against the convention drive. They argue that Brown, a potential rival for the presidency, erred badly in boarding this bandwagon, and that Carter can capitalize by fighting it. And they think such an effort would win back Democratic liberals who are disenchanted with Carter.

The burgeoning backlash from Washington comes just as the convention drive is moving into already hostile territory. Since most states in the South and West already have passed resolutions, the fight for the last halfdozen states will have to focus on the industrial northeast -- where local budgets are particularly dependent on federal money.

The proponents recently have suffered the first serious setbacks of their four-year effort. Last week's defeat of a resolution in Brown's own state, California, was a significant psychological blow. Although the Taxpayers' Union lists Iowa as one of the states that passed a resolution, that legislature two weeks ago defeated a call for an immediate convention, approving in its place a request for a convention only if Congress fails to act on the budget issue by July 1, 1980.

If more states do demand a convention on the budget issue, Congress will begin to scrutinize the resolutions it has received -- and that could be fatal to the convention drive.

Of the 28 resolutions approved to date, 16 ask Congress either to convene a constitutional convention or to propose an amendment of its own. Congressional lawyers studying the issue say flatly that such conditional requests are not valid demands for a convention. Some of those 16 set no time limit for Congress to propose an amendment -- so there is no date when the "conditional" convention call becomes effective.

Although the intent of most of the 28 resolutions is clear, the states have passed a hodgepodge of different proposals. In some cases, different houses of the same legislature sent in different resolutions. A dozen states that passed resolutions have not sent them to Congress.

There also are technical difficulties in some state resolutions. Delaware, for example, asked for a convention only if 33 other states propose an amendment identical to Delaware's version. To date, no other state has adopted Delaware's wording.

Backers of the convention drive say Congress would be guilty of outrageous nitpicking if it rejected a convention on these technical grounds. Dave Keating of the Taxpayers' Union agrees that some resolutions are "marginal," but says it is unrealistic to expect 34 different legislatures to agree on nearly identical language.

Keating says any resolution rejected by Congress would be revised and resubmitted quickly. But Fred Wertheimer, a vice president of Common Cause, which opposes a convention, says this is not the case. "A lot of states that passed this back when nobody was looking very hard at it might think twice if they got another chance."

In any case, there is a broad and bipartisan consensus on Capitol Hill that Congress would not call a constitutional convention even if 34 arguably valid resolutions were submitted.

"Everybody here is afraid of the rogue elephant idea of a convention," said one senior House Republican. "So we would give them something, a balanced budget next year or maybe even a proposed amendment, before we would let the states get into the amending business."