Preheat the oven to 325°F. Line an 8-inch springform pan with parchment and grease the sides.
Whisk the almond flour, coconut flour, baking powder, and salt in a bowl.
Separate 2 of the eggs, placing the whites in a clean bowl. Whisk the remaining 2 whole eggs, 2 egg yolks, allulose, Greek yogurt, avocado oil, vanilla extract, and lemon zest until smooth.
Stir the dry ingredients into the yogurt mixture. Beat the 2 egg whites until soft peaks form, then gently fold them into the batter.
Spread the batter into the pan and dot the top with blackberries. Bake for 32 to 38 minutes, until puffed and lightly golden.
Cool completely in the pan. Whip the heavy cream, powdered allulose, and vanilla extract until soft peaks form.
Serve the cake with the vanilla cream spooned over each slice.