Preheat the oven to 350 degrees F (175 degrees C). Coat a 9x13-inch baking pan with cooking spray.

Combine gingersnap crumbs and melted butter in a small bowl until well incorporated. Press evenly and firmly into the bottom of the prepared baking pan.

Bake in the preheated oven until crust is browned and slightly firm, 6 to 8 minutes; crust will further set and harden as it cools, so do not overcook. Remove from oven and let cool completely; leave oven on.

Meanwhile, combine pumpkin, flour, brown sugar, and pumpkin pie spice in a medium-sized bowl until well combined.

Beat cream cheese, white sugar, and vanilla extract together in a mixing bowl using an electric mixer on medium speed until smooth and creamy. Add eggs 1 at a time, combining thoroughly after each addition.

Measure 1 cup of cream cheese mixture and add to pumpkin mixture, hand stirring until well combined. Spread remaining cream cheese mixture evenly over the cooled crust.

Drop rounded tablespoons of pumpkin mixture randomly over the entire surface of the cream cheese layer. Use the handle of a wooden spoon to carefully swirl the 2 batters together into a marbled effect.

Bake in the hot oven until center is set and a toothpick inserted in the center comes out fairly clean, 30 to 35 minutes. Cool completely at room temperature until thoroughly set, about 1 hour.

Cut into 24 bars. Refrigerate any leftover bars.