A sensitive game of cat-and-mouse involving evangelical Christians, Congress, the Constitution, national politics and school prayer took an abrupt turn yesterday.

After months of resisting, Rep. Robert W. Kastenmeier (D-Wis.) announced that his House Judiciary subcommittee will hold hearings this month on the volatile school-prayer issue.

Kastenmeier, in effect, blocked at the pass a rapidly growing movement by the evangelicals to force pro-prayer legislation in the House floor for a vote before Congress adjourns in October.

"This is the worst thing that could happen to us. They have cut us off from the floor," said Bill Chasey, chief lobbyist for the evangelicals and orchestrator of the drive to circumvent Kastenmeier's subcommittee. "They'll delay, I know they'll delay."

The issue that has been bitterly debated behind the scenes for months involves legislation, deemed unconstitutional by some scholars, that would end federal court jurisdiction in any school-prayer cases.

Several Supreme Court rulings have held against organized prayer in public schools as a First Amendment infringement. The new legislation would not require prayer; it would prevent federal courts from regulating it.

The legislation started out in the Senate under the sponsorship of Sen. Jesse A. Helms (R-N.C.). He got it attached to another court bill.

The senate passed the bill in April 1979 and sent it to the House, where it was expected to go to quiet interment, largely on the grounds that it was without constitutional merit.

Kastenmeier bottled up the bill in his courts and civil liberties subcommittee. But apparently neither he nor the Senate counted on the evangelical movement to react as strenuously as it has.

Spurred on by the Rev. Jerry Falwell of Lynchburg, Va., and other religious broadcasters whose audiences are said to be in the many millions, the evangelicals turned on Capitol Hill. Falwell's Moral Majority organization is playing a lead role in the lobbying.

They enlisted Rep. Philip M. Crane (R-Ill.) to circulate a petition that would force the bill out of the subcommittee and send it to a vote on the House floor.

In an election year in which the evangelicals have become increasingly political, Crane found quick support for his petition -- so far, about 180 of the 218 names needed to get the bill to the floor.

Kastenmeier's announcement of hearings on July 29 and 30 was calculated to feeling the evangelicals' pressure, from signing the petition.

"They'll have hearings, then it will have to go to the full committee. Then we will be out of time and Congress will be gone," said Chasey, the lobbyist from Roundtable Issues and Answers who is helping Crane get names on the petition.

The evangelicals' hope, Chasey and others explained, is to get the legislation through Congress and to put the Carter White House in a ticklish political dilemma over signing or vetoing it.

The Carter administration is strenuously opposed to the Helms amendment, saying it is unconstitutional and a precedent for letting emotion overrule federal law on other issues.

The administration is supported on this issue by most of the country's major religious denominations, represented by the National Council of the Churches of Christ. The U.S. Catholic Conference has not taken a position on the Helms amendment.

In a letter last week to all House members, the council urged against signing the Crane petition and came out against the amendment itself.

The letter said removing federal court jurisdiction in school prayer cases "would deny protection to the religious liberties of both those students who have a preference for or commitment to a particular form of worship and those who do not practice any religion."

"The Supreme Court has never denied children the right to pray or to read the Bible in school," the letter said. "It has however, said the state may not impose, sanction or encourage such an exercise."

But those arguments do not deter Chasey and the "pro-family and pro-normal values" for which he lobbies.

"The Gallup Poll shows that 76 percent of the American people want voluntary school prayer. . . . We have the numbers, we have the political shift to the right. Ronald Reagan supports our amendment. We could take over the House and Senate. Every member of Congress is being made aware of this," he said.