KANSAS CITY, Mo.—The Show Me state is headed for a showdown over a move to rein in the adult-entertainment industry at a time when every job counts—even those of strippers.

Last month, the Republican-controlled legislature passed one of the nation's toughest state laws aimed at strip clubs and other adult-entertainment venues. It would ban nude dancing and the serving of alcohol in adult cabarets, force strip clubs to close at midnight and forbid seminude dancers to touch patrons.

Missouri...