The annual deficit reached $1.375 trillion last year, up from $983 billion in 2019, the last year before the Covid-19 pandemic prompted vast relief spending, and is projected to double in the next decade. Even aside from the linkage with the debt ceiling, the two sides are drastically apart on how to address the red ink. Mr. Biden has proposed a budget that would reduce projected deficits by nearly $3 trillion over 10 years by increasing taxes on corporations and the wealthy, while Mr. McCarthy’s plan would scale back deficits by $4.8 trillion over a decade largely through cuts in discretionary programs.

In speaking to a swing-voting New York suburb, Mr. Biden seemed to have two audiences — voters outside the capital who may not be paying as much attention to the debate and Mr. Lawler. A 36-year-old former political operative and first-term Republican, Mr. Lawler is an obvious target for the White House to try to sway. He ousted Representative Sean Patrick Maloney, then the chairman of House Democrats’ campaign operation, in a district that Mr. Biden won by 10 percentage points.

In Washington, Mr. Lawler has positioned himself as a serious-minded moderate, breaking with his party on some cultural issues while supporting Mr. McCarthy’s debt ceiling and spending proposal. Both parties view him as one of the most vulnerable Republicans in 2024, and Democrats are already lining up millions of dollars and potential candidates to defeat him.

For now, Mr. Lawler appears to be toeing a careful line between his party’s leaders and the president. When the White House reached out with an invitation to the event that many in the G.O.P. would have shunned, he promptly accepted. In media interviews before and after the speech, Mr. Lawler reiterated he would not support a default. But he also chastised Mr. Biden for not engaging with Mr. McCarthy sooner and insisted on broad spending cuts.