Adjust oven rack to middle position and heat oven to 350 degrees. Grease two 9-inch-wide by 2-inch-high round cake pans and line bottoms with parchment paper. Grease paper rounds, dust pans with flour, and knock out excess. Whisk flour, baking powder, baking soda, salt, and 1 1/2 cups sugar together in large bowl. In 4-cup liquid measuring cup or medium bowl, whisk together melted butter, buttermilk, oil, vanilla, and yolks.
In clean bowl of stand mixer fitted with whisk attachment, beat egg whites at medium-high speed until foamy, about 30 seconds. With machine running, gradually add remaining 1/4 cup sugar; continue to beat until stiff peaks just form, 30 to 60 seconds (whites should hold peak but mixture should appear moist). Transfer to bowl and set aside.
Add flour mixture to now-empty mixing bowl fitted with whisk attachment. With mixer running at low speed, gradually pour in butter mixture and mix until almost incorporated (a few streaks of dry flour will remain), about 15 seconds. Stop mixer and scrape whisk and sides of bowl. Return mixer to medium-low speed and beat until smooth and fully incorporated, 10 to 15 seconds.
Using rubber spatula, stir 1/3 of whites into batter to lighten, then add remaining whites and gently fold into batter until no white streaks remain. Divide batter evenly between prepared cake pans. Lightly tap pans against counter 2 or 3 times to dislodge any large air bubbles.
Bake until cake layers begin to pull away from sides of pans and toothpick inserted into center comes out clean, 20 to 22 minutes. Cool cakes in pans on wire rack for 10 minutes. Loosen cakes from sides of pans with small knife, then invert onto greased wire rack and peel off parchment. Invert cakes again and cool completely on rack, about 1 1/2 hours.
Once the cake has cooled, spread half the pudding on the top of the first layer then cover with sliced bananas. Spread the rest of the pudding over the bananas, being careful to not disturb them too much. Stack the second layer on top of the first and frost the entire cake with whipped cream. Decorate the top with more sliced bananas.