Preheat oven to 325°.
To prepare crust, combine crumbs and egg white, tossing with a fork until moist. Press into bottom of an 8-inch springform pan coated with cooking spray. Bake at 325° for 5 minutes or until lightly browned; cool on a wire rack.
To prepare filling, place cottage cheese in a food processor; process until smooth (about 2 minutes), scraping the sides of bowl once. Add cream cheese, and process until smooth. Add 2/3 cup sugar, 2 tablespoons cornstarch, vanilla, and egg; pulse just until combined. Pour batter into prepared pan.
Bake at 325° for 30 minutes. Reduce oven temperature to 300° (do not remove cheesecake from oven); bake an additional 20 minutes or until center of cheesecake barely moves when pan is touched. Remove cheesecake from oven; run a knife around outside edge. Cool to room temperature. Cover and chill at least 8 hours.
To prepare syrup, combine dark rum and raisins in a microwave-safe bowl; microwave at high 2 minutes. Cover with plastic wrap; let stand 10 minutes.
Combine 1/4 cup sugar and 1 tablespoon water in a small saucepan over medium heat; cook, without stirring, 5 minutes or until golden. Combine 1 teaspoon cornstarch and 1 tablespoon water, stirring with a whisk. Stir rum mixture and cornstarch mixture into sugar mixture; bring to a boil. Cook 1 minute. Serve syrup chilled or at room temperature with cheesecake.