New Jersey Senate President Steve Sweeney, who has feuded with public-sector unions in his state for years, makes it a point to tout his dedication to the broader labor movement.

“I’m a vice president of the Ironworkers International of North America,” Mr. Sweeney, a Democrat, said in an interview last year with The Wall Street Journal. “There is no one you are going to find that is more union than me.”

Fellow ironworkers across the Hudson River in New York disagree.

Mr. Sweeney, the top-ranking state legislator in New Jersey, is in a quarrel with members of his own union after a labor dispute earlier this year at the $25 billion Hudson Yards project on Manhattan’s westside.

The fight comes as Mr. Sweeney continues to inflame public unions in New Jersey by pushing for spending cuts to the pension and health-care benefits of unionized state employees. In 2011, Mr. Sweeney supported then Gov. Chris Christie, a Republican, and his efforts to curb pension costs.