Asked about Ms. Rutledge’s criticism, Ms. Sanders ignored her rival and trumpeted her own record-setting early fund-raising. “I take nothing for granted,” she said via text message.

Should Ms. Sanders emerge as the Republican standard-bearer, she may face a third-party opponent from well outside the pro-Trump orbit. State Senator Jim Hendren, who left the G.O.P. after the Jan. 6 riot, and Davy Carter, a former state House speaker, are both considering bids.

In separate interviews, they said they would not compete with one another in the same race. “I’m convinced that even in Arkansas, Trump and Trumpism is a slow-sinking ship,” said Mr. Carter, who as speaker helped push through Medicaid expansion. He said that a successful challenge to Trumpism would not happen unless liberals, moderates and anti-Trump Republicans “organize in one lane.”

Asked who he’d ultimately back in the governor’s race, Mr. Hutchinson said, “I expect to support the Republican nominee.”

But he acknowledged talking extensively with his nephew, Mr. Hendren, saying they share “the same frustrations” about the party, except that Mr. Hutchinson is determined to fight from within the tent. Offering some barely veiled advice for Ms. Sanders, he said: “Leadership is about bringing people along and not giving in to a lie.”

The governor, and most observers, are deeply skeptical that an independent could win statewide. Indeed, more than a year and a half before Ms. Sanders would even take office, many insiders have moved on to discussing what sort of governor she would be.

Would she repurpose Mr. Trump’s media-bashing and grievance-oriented politics to stay in the national headlines, and perhaps propel a presidential run of her own, or would she mirror her father’s more pragmatic approach to the office? While he is now known for his own Fox News and social media profile, Mr. Huckabee governed in the political center, even incurring the wrath of the far right, whom he labeled “Shiite conservatives.”