Preheat oven to 350 degrees F. Spray an 8x8-inch square baking pan with nonstick spray, line with parchment paper and spray parchment.
In a medium bowl, whisk to combine the flour, baking powder and salt. In a stand mixer fitted with the paddle attachment, beat the butter, granulated sugar and light brown sugar on medium speed until well blended. Beat in eggs, one at a time, then beat in the bananas. Reduce the speed to low and add the flour mixture, stirring until just combined. Stir in the vanilla extract and chocolate chips.
Transfer batter to the prepared baking dish and spread to make smooth and even. Sprinkle 1/4 cup sugar over the top, then bake until golden and a toothpick inserted in the center comes out clean, about 35 minutes. Transfer to a wire rack to cool completely.
Use a serrated knife and carefully slice the banana cake in half horizontally. Remove the top half and set aside. Spread the ice cream in a smooth layer over the cake, then top with the caramel sauce. Place the top back on the bread and run a knife flat along the edge to make smooth.
Insert 8 popsicle sticks into the ice cream, 4 evenly spaced on opposite sides. Cover with plastic and freeze until set, at least 6 hours and up to overnight.
In a medium bowl, combine the white chocolate chips and oil. Melt in the microwave or over a double boiler. Let cool to room temperature.
Line a baking sheet with wax paper. Remove the cake from the freezer and cut into 8 equal sandwiches. Holding the popsicle sticks, dip the edge of the sandwich on a diagonal into the chocolate, coating from the top corner all the way to the bottom corner. Top the chocolate with sprinkles. Place on a wax lined baking sheet and return to freezer until ready to serve.