To prepare cake, coat a 10-inch tube pan with cooking spray; dust with breadcrumbs. Set aside.
Lightly spoon flour into dry measuring cups; level with a knife. Combine flour, baking powder, baking soda, and salt, stirring well with a whisk. Place butter and granulated sugar in a large bowl; beat with a mixer at medium speed until light and fluffy. Add eggs, 1 at a time, beating well after each addition. Add milk, 1 tablespoon rum, and vanilla; beat until combined. Beating at low speed, add flour mixture and sour cream alternately to sugar mixture, beginning and ending with flour mixture; beat until just combined.
Spoon batter into prepared pan. Bake at 350° for 1 hour or until a wooden pick inserted in center comes out clean. Cool in pan 10 minutes. Loosen cake from sides of pan using a narrow metal spatula. Place a plate upside down on top of cake; invert onto plate. Invert cake again. Pierce cake liberally with a wooden pick.
While cake bakes, prepare glaze. Combine brown sugar, 2 tablespoons rum, and water in a small saucepan; bring to a boil, stirring until sugar dissolves. Add 1 1/2 tablespoons butter, stirring until butter melts. Drizzle half of warm glaze evenly over warm cake; allow mixture to absorb into cake. Drizzle remaining glaze over cake. Cool cake completely.