Mr. Trump was supposed to shatter their hold on power.

That was why many activists and voters swallowed their own reservations about the sincerity of his commitment to their causes, and brushed aside concerns from many fellow Republicans that they were cutting a deal with a charlatan who would inevitably sell them out. Seeing no other alternative but Hillary Clinton as president, they bought into Mr. Trump’s “drain the swamp” promises to upend Washington and forged a bond over their mutual contempt with the Republican Party establishment.

But on Wednesday, prominent conservatives scoffed at the deal that Mr. Trump signed onto — announced first, no less, by congressional Democrats — as something straight from the swamp.

“I know for certain,” said Jenny Beth Martin, a founder of Tea Party Patriots, that grass-roots conservatives “did not work so hard last year to elect majorities in the House and the Senate and get Trump elected in the White House to enact liberal policy priorities.”

With his approval ratings languishing at historic lows, Mr. Trump can hardly afford to lose much more support. And the Republican Party, which was already anxious about losing control of the House in next year’s midterm elections, cannot remain competitive against Democrats if many of its most motivated and reliable voters stay home because they believe their president has betrayed them.

Republicans were already facing a to-do list that threatened to drive them apart over the next several months.

Besides another round in December on the debt ceiling and keeping the government funded, there is a continuing debate over what kinds of tax cuts Congress should approve and whether, as many conservative activists and Democrats believe, the plan Mr. Trump supports is too generous to corporations at the expense of individuals. Also to be resolved are questions about whether the president’s border wall will be paid for and if he can deliver the deep cuts to the federal bureaucracy he promised.