Ronna McDaniel, Mr. Trump’s handpicked chairwoman, has secured the president’s support for her re-election to another term in January, when the party is expected to gather for its winter meeting. But her intention to run with Mr. Trump’s blessing has incited a behind-the-scenes proxy battle, dividing Republicans between those who believe the national party should not be a political subsidiary of the outgoing president and others happy for Mr. Trump to remain in control of it.

While many Republicans are hesitant to openly criticize their president at a moment when he is refusing to admit he has lost, the debate crystallizes the larger question about the party’s identity and whether it will operate as a vessel for Mr. Trump’s ambitions to run again in four years…

The continuing influence of Mr. Trump could also have implications for some of the national committee’s most critical assets: Its voter data and donors lists contain thousands of names of contributors and detailed information about supporters. The voter data in particular is a focus of attention, after distrust arose between the committee and the Trump campaign over the data’s use in the final months of the campaign.