LAS VEGAS—The neon billboard atop the Best Western Plus Casino Royale doesn’t hawk big-name entertainers, mouthwatering seafood or pool parties with bottle service like most resorts along the Strip.

For 10 days every December, it’s one of the toughest tickets in town. The National Finals Rodeo draws an average of 170,000 people to the valley each year. Only about 17,500 can be packed into UNLV’s ...