Preheat oven to 350°F. Butter and flour 3 (9-inch) round cake pans. Line bottom of pans with parchment paper; butter and flour pans again.
In the bowl of a stand mixer fitted with the paddle attachment, beat butter and sugar at medium speed until fluffy, 3 to 4 minutes, stopping to scrape sides of bowl. Add eggs, one at a time, beating well after each addition.
In a medium bowl, whisk together flour, baking powder, baking soda, and salt. In a small bowl, stir together coconut milk and buttermilk. With mixer on low speed, gradually add flour mixture to butter mixture alternately with coconut milk mixture, beginning and ending with flour mixture, beating just until combined after each addition. Beat in vanilla. Divide batter among prepared pans.
Bake until a wooden pick inserted in center comes out clean, about 30 minutes. Let cool in pans for 10 minutes. Remove from pans, and let cool completely on wire racks. Brush layers with Peach Simple Syrup.