ANCHORAGE—The nomination fight over Supreme Court nominee Brett Kavanaugh is testing the independent streaks of several moderate senators, including Alaska’s Lisa Murkowski whose extraordinary, write-in re-election victory eight years ago has left her less shackled to the party hierarchy.

Ms. Murkowski this week dismissed demands to act quickly on the nomination. Senate Majority Leader Mitch McConnell (R., Ky.) has said that he plans to vote this week, nearly in tandem with the expected release of a Federal Bureau of Investigation probe into allegations of sexual assault and misconduct leveled against the nominee.

“He talked about a vote last week, too,” Ms. Murkowski said.

To prevail, Mr. McConnell can lose only one Republican if all Democrats oppose the nominee, who has denied the allegations against him. Three Republicans are undecided: Ms. Murkowski as well as Sens. Jeff Flake of Arizona and Susan Collins of Maine.

While Ms. Collins and Mr. Flake have been in the spotlight, particularly because of his decision to broker a Judiciary Committee deal to allow time for the FBI probe, Ms. Murkowski has been able to minimize attention in Washington.