Mr. Pence insisted that he and Mr. Trump were trying to belittle President Obama rather than to laud Mr. Putin.

The very nature of Mr. Pence’s visit to Capitol Hill offered some insights as to why Mr. Trump, who had his own testy visit with Senate Republicans here in July, is still confronting difficulties with his party. Playing the part of troubleshooting diplomat as much as running mate, Mr. Pence spent much of the day behind closed doors reassuring his former colleagues, with whom he retains close ties, that they could trust Mr. Trump. Mr. Pence said Mr. Trump behaved differently in private, and even had a spiritual side.

Congressional Republicans said they were pleased that the presidential race was tightening and that Mr. Trump appeared more viable, partly because their own prospects depend on his being competitive enough that Republicans still bother to vote.

With his modest uptick in the polls, Mr. Trump may have forestalled an outright break with Republicans in Congress. As recently as a few weeks ago, party leaders and strategists in Washington were preparing to take emergency measures, having their candidates disavow Mr. Trump and present themselves to voters as a counterweight to a President Clinton.

Republicans may still activate those plans if Mr. Trump stumbles badly against Mrs. Clinton in the final stretch of the race. But for now, the party has adopted a middle-ground approach, with most Republicans in Washington nominally supporting Mr. Trump while advancing campaign messages that differ widely from his in almost every respect.

That distance was on display when congressional Republicans refrained from echoing Mr. Trump’s indictment of Mrs. Clinton over her saying that half his supporters were bigots, sexists and homophobes.

Few Republicans want to confront thorny follow-up questions about Mr. Trump’s supporters, and fewer still want to make matters of race, gender and sexual orientation central to their campaigns. The silence about what has for the last three days been at the core of Mr. Trump’s campaign also underscored a more fundamental gap: Republicans on Capitol Hill have their own agenda and intend to run their campaigns apart from Mr. Trump — not as a unified ticket.