Preheat the oven to 350°F (180°C). Grease a 13 x 18-inch (33 x 45 cm) rimmed baking sheet with nonstick spray and line with parchment paper.
In a medium bowl, whisk together the all-purpose flour, cake flour, baking soda, baking powder, and salt until combined.
Add the butter and granulated sugar to a large bowl and cream with an electric hand mixer on medium-high speed until light and fluffy, about 3 minutes. Reduce the mixer speed to medium and beat in the eggs, 1 at a time, scraping down the sides of the bowl as needed. Add the vanilla and beat until incorporated.
Add a third of the dry ingredients, followed by half of the buttermilk, another third of the dry ingredients, remaining buttermilk, and remaining dry ingredients, beating well after each addition. Fold in the rainbow sprinkles.
Pour the batter onto the prepared baking sheet and use an offset spatula to spread evenly.
Bake until the cake is light golden on top and a toothpick inserted in the center comes out clean, 18–22 minutes. Set the baking sheet on a wire rack and let the cake cool completely before frosting.
While the cake is cooling, make the buttercream frosting: In the bowl of a stand mixer fitted with the paddle attachment (or in a large bowl with an electric hand mixer), beat the butter on medium-high speed until creamy, about 2 minutes. Add the powdered sugar and beat on low speed for 30 seconds to incorporate. Increase the mixer speed to medium-high and beat for 3 minutes, stopping as necessary to scrape down the sides of the bowl, until fluffy. Add the heavy cream, vanilla, and salt and beat on medium speed for another 2 minutes, or until the buttercream is no longer grainy. If desired, continue mixing for another 1–2 minutes for a fluffier consistency.
Assemble the cake: Use the parchment to lift the cake from the pan onto a flat surface. Using a sharp knife, cut the cake in half crosswise. Spread about half of the buttercream across one half of the cake. Place the other cake half on top of the buttercream, top down, and gently press down to adhere. Peel off the parchment lining. Spread the remaining buttercream evenly over the top layer of cake, reserving some buttercream for coloring, if desired. Decorate with colored buttercream, more sprinkles, or edible flowers as desired.
Chill the cake in the refrigerator for at least 30 minutes, or until the buttercream is firm. Using a very sharp knife, trim the edges of the sheet cake for clean lines. Slice crosswise into 8 strips, then slice in half vertically for 16 bars, running the knife under warm water and drying between cuts if desired for cleaner lines.
Store the cake bars in an airtight container in the refrigerator for up to 1 week, or in the freezer for up to 3 weeks.