In the run-up to the Academy Awards, Neon took a unique approach to scoring the season’s top prize for “Anora.” While other studios spent their money on tried-and-true strategies, from tastemaker screenings to glossy ads, Neon pulled up a tow truck in front of a Los Angeles auto body shop for a pop-up “Anora” merchandise sale in November — complete with branded T-shirts and thongs. Some 300 fans queued up before the vehicle even arrived. And for its first screening of the film about a Brooklyn escort who marries a Russian oligarch’s kid, the boutique distributor filled the audience with sex workers rather than Oscar voters.

“We follow the beat of our own drum,” says Tom Quinn, Neon’s CEO. “The idea of pandering to the campaign as opposed to being who you are as a film is a big, stark difference. We never play to the campaign. We always play to the film,...