Only hours before the gathering began, the news broke that Ms. Warren, now seen as Mr. Biden’s chief rival, had out-raised him in the last three months by nearly $10 million — $24.6 million to $15.2 million. He lagged behind Senator Bernie Sanders ($25.3 million) and Mayor Pete Buttigieg ($19.1 million), too.

While some were downbeat about the financial figures, Mr. Biden’s slip in the polls and the onslaught from Mr. Trump, the Biden loyalists described a sense of almost burden and responsibility. If Mr. Biden is to compete financially with the online windfalls of Mr. Sanders and Ms. Warren, several donors said, they believe it would be on the shoulders of the bundlers assembled here.

“All of us realize that Joe Biden does not have the online fund-raising capability of a Warren. Warren has been doing it longer than him. Sanders has been doing it longer than him,” said Dick Harpootlian, a prominent South Carolina supporter who hosted a fund-raiser for Mr. Biden in May.