Grease bottom and sides of 8-inch pan. Stir flour, pudding mix, and baking powder together in a large mixing bowl. Add the 1/2 cup milk, the butter, and egg. Beat with mixer on medium for about 2 minutes. Pour batter into prepared pan. Slice bananas, and arrange slices atop the batter and set aside.
Beat cream cheese, sugar, 2 tablespoons of milk, lime juice, and rum extract in a medium bowl with mixer on medium till smooth. Spoon cheese mixture over banana slices, carefully spreading mixture over them.
Bake at 350°F for 30-35 minutes or until center is set. Let set about half an hour until cool and then refrigerate for 2 hours before serving.