While the cake is cooling in the fridge, prepare the buttercream icing. In a stand mixer or large bowl, beat together butter, powdered sugar, and half of the remaining pink champagne. You want to add just enough champagne to make the frosting about the texture of soft butter, so start with a small amount and add more only if needed. The frosting should be fluffy, with firm peaks. Tint frosting to your desired shade of pink.