Preheat oven to 350 degrees F. Coat cake pan with nonstick spray; set aside. Cook brown sugar and butter in a small deep heavy saucepan over medium heat, whisking occasionally, until mixture is bubbling and sugar is completely dissolved. Boil, whisking occasionally, until caramel darkens, thickens, emulsifies, and pulls away from sides of pan, about 3 minutes.
Remove from heat and whisk in bourbon (mixture will bubble vigorously). Return caramel to a boil; cook for 1 minute. Pour caramel into prepared cake pan and swirl pan until bottom is completely covered. Let cool completely before proceeding (caramel will harden).
Cut skin from pineapple, making sure to remove all brown eyes. Cut 1/4-inch–1/3-inch-thick rounds from pineapple. Using biscuit cutter, remove core to form rings. (If pineapple is very juicy, drain rings briefly on paper towels.) Arrange pineapple rings decoratively on top of caramel in cake pan without overlapping. (Cut rings into smaller pieces to fill any large gaps, if desired.).
Whisk flour, baking powder, salt, and cardamom in a medium bowl. Using an electric mixer, beat sugar and butter in another medium bowl until light and fluffy, 5–6 minutes. Beat in vanilla. Add eggs one at a time, beating to blend between additions. Scrape down sides of bowl. Beat egg mixture until doubled in volume and very light and fluffy, about 2 minutes. Scrape bottom of bowl well, making sure to incorporate all butter into eggs. Beat for 1 minute longer.
Beat in flour mixture in 3 additions, alternating with sour cream in 2 additions, beginning and ending with dry ingredients. Scrape batter into pan; smooth top.
Bake cake until a tester inserted into the center comes out clean, 50–60 minutes. Transfer to a wire rack. Let cake cool in pan for 30 minutes. Run a sharp knife around sides of pan to loosen cake; tap bottom of pan and invert cake onto a large plate.