Preheat your oven to 350 F and prepare three 8-inch round cake pans with grease and parchment paper. Whisk together the almond flour, tapioca starch, baking powder, baking soda, salt, and finely ground lavender.
Beat the butter and sugar together on medium-high speed for 4 to 5 minutes until pale and fluffy. Add the eggs one at a time, beating well after each, then mix in the vanilla. Add the flour mixture in three additions alternating with the buttermilk, mixing on low until just combined.
Add a tiny drop of purple food coloring if desired for a subtle lavender hue. Divide the batter among the pans and bake for 22 to 26 minutes until a toothpick comes out clean. Cool in the pans for 10 minutes, then turn out onto racks.
Make the blackberry compote by combining the blackberries, sugar, and lemon juice in a small saucepan over medium heat. Cook for 6 to 8 minutes, stirring and gently pressing the berries, until they break down and release their juices.
Stir in the tapioca starch mixture and cook for another minute until thickened. Strain through a fine mesh sieve to remove the seeds, pressing to extract all the liquid. Let cool completely.
For the buttercream, place the egg whites and sugar in a heatproof bowl over simmering water. Whisk constantly until the sugar fully dissolves and the mixture reaches 160 F.
Remove from heat and beat on high speed with an electric mixer for 8 to 10 minutes until stiff, glossy, and completely cooled. Add the softened butter a few cubes at a time, beating on medium speed. It may look curdled briefly but keep beating until it comes together smooth. Mix in the vanilla and lavender extract. Tint a soft purple with gel food coloring.
Level the cake layers if needed. Place the first layer on a cake stand and pipe a ring of buttercream around the edge to create a dam. Spoon half the blackberry compote inside the dam and spread evenly. Place the second layer on top and repeat the dam and compote. Top with the third layer.
Crumb coat the entire cake with a thin layer of buttercream and chill for 20 minutes. Apply the final coat of buttercream, smoothing it with a bench scraper. Use any leftover buttercream to pipe rosettes or swirls on top if desired.
Decorate the top with fresh blackberries and dried lavender sprigs. Refrigerate for at least 30 minutes before slicing. Bring to room temperature for 15 minutes before serving for the softest texture.