For the pizza dough: Whisk together the flour, baking powder and salt in a medium bowl.
Beat together the sugar and butter on medium-low speed in a stand mixer fitted with a paddle attachment until light and creamy. Beat in the lemon zest, vanilla and egg. Increase the speed to medium and add the flour mixture in 2 batches, scraping down the sides of the bowl as necessary, and beat until just incorporated.
Divide the dough in half and pat each piece into a disk. Wrap separately with plastic wrap and freeze for 45 minutes.
Preheat the oven to 350 degrees F. Line a baking sheet with a silicone baking mat or parchment and dust it with flour. Roll one of the discs into a round that is approximately 1/4 inch thick and 6 1/2 inches across. Put a plate or inverted bowl of a similar size on top of the dough and cut off any excess dough to create a clean edge. Transfer to a lightly-floured work surface.
Dust the baking sheet again with flour and repeat with the second disc. Brush off the baking sheet and arrange the 2 dough rounds on it. Bake until lightly browned around the edges, about 15 minutes. Cool completely on the baking sheet.
Meanwhile, make the frosting: Beat together the cream cheese, yogurt and honey on medium speed in a stand mixer fitted with a paddle attachment until thick and smooth. Mash the blackberries in a small bowl. Strain through a fine mesh sieve; you should have about 2 tablespoons of juice (see Cook's Note). Add the juice to the cream cheese mixture and beat until the frosting is purple with no white streaks. Frost the 2 cooled "pizza" rounds. Refrigerate until ready to serve.
Decorate each pizza with blueberries, raspberries and blackberries. Sprinkle each with coconut. Slice each pizza into 4 pieces.
Add more blackberry juice for a more intense purple colored frosting.