Arrange a rack in center of oven; preheat to 350°. Line a 13" x 9" baking dish with parchment, leaving an overhang on 2 opposite long sides. Grease parchment with cooking spray.

In a food processor, pulse gingersnap cookies until fine crumbs form, about 45 seconds. Add butter, sugar, pumpkin pie spice, and salt and pulse until mixture resembles wet sand, about 30 seconds more. Pour into prepared dish and pat down with your hands or the back of a measuring cup.

Bake crust until deeply golden brown, 10 to 12 minutes. Transfer to a wire rack and let cool slightly. Reduce oven temperature to 325°.

In the large bowl of a stand mixer fitted with the whisk attachment (or using a handheld mixer), beat cream cheese on medium-high speed until smooth. Add sugar, sour cream, and vanilla and beat until smooth. Scrape down bowl. Add eggs, one at a time, beating to blend after each addition.

Spread pumpkin puree on a large plate, then pat as dry as you can with paper towels. Transfer pumpkin puree to a medium bowl. Scoop ¾ cup cheesecake batter into bowl with pumpkin puree. Add pumpkin pie spice and whisk until incorporated.

Pour remaining cheesecake batter over crust in an even layer. Dollop pumpkin batter in about 8 scoops across top of cheesecake batter. Using the back of a fork or a spoon, swirl batters 6 to 7 times to create desired pattern.

Bake bar until top is dry, edges are slightly browned, and center slightly jiggles, about 30 minutes. Transfer to a wire rack and let cool 10 minutes. Refrigerate until set, at least 1 hour and up to 24.

Using parchment overhang, lift out bar and transfer to a cutting board. Slice into squares.