Preheat oven to 350 degrees F (175 degrees C). Grease and line the bottom of five 8-inch cake pans with parchment paper.
Beat butter and white sugar together in the bowl of a stand mixer fitted with the paddle attachment on medium-low speed until creamy and light, 3 to 5 minutes. Turn the mixer to low speed and beat in the egg whites. Turn off mixer and scrape the sides of the bowl with a rubber spatula. Continue beating until combined.
Mix flour, 2 tablespoons plus 1 teaspoon baking powder, and salt into creamed butter mixture on low speed. Add Champagne, sprinkles, and vanilla extract; beat until batter is just combined. Scrape down the bowl with the spatula. Divide batter among the prepared pans.
Bake in the preheated oven until cakes are golden around the edges and centers feel springy when gently pressed, 20 to 25 minutes. Cool cakes in the pans on a wire rack for 10 minutes. Invert pans to release cakes and cool completely, at least 1 hour.
Cut each cake in half horizontally, creating 10 rounds.
Line two 8-inch cake pans with parchment paper in the bottom. Cut two 6-inch wide strip of parchment paper and the the sides of each pan.
Place 1 cake layer in each pan and top each with a thin layer of hot fudge. Top fudge layer with a layer of ice cream. Top ice cream with a cake layer. Spread caramel over the cake layer and top with a layer of ice cream and finish with a cake layer. Freeze for 1 hour.
Top the layered cakes with marshmallow cream and another layer of ice cream. Repeat layering with marshmallow cream and ice cream, alternating with hot fudge and caramel, until all the cake layers have been used. Freeze cakes until completely solid, at least 2 hours.
Carefully remove cakes from the pans. Add 1 final layer of ice cream to 1 cake and top with the other cake.
Beat cream in a chilled glass or metal bowl with an electric mixer until stiff peaks form. Spread whipped cream over entire cake; freeze until set, at least 30 minutes. Drizzle caramel over cake before serving.