Preheat oven to 350 degrees. Line a 9 x 9 inch square baking pan with parchment.
With a hand or stand mixer, whip coconut oil, sugar, eggs, vanilla, and strawberry puree until combined. Measure in flour, baking soda, baking powder, and salt. Beat until just combined. Do not overmix.
In a separate bowl, beat cream cheese, sugar, and flour for about 3 minutes until fluffy. Add eggs one at a time. Blend in vanilla until incorporated.
Pour half of the strawberry cake batter into the prepared pan. Top with half of the cheesecake batter. Dollop remaining cake batter atop the cheesecake layer, then pour leftover cheesecake filling. Bake 40-45 minutes, tenting the pan halfway through with foil to prevent browning, until the center is set.
Cool cake in the pan for 15 minutes, then transfer to the freezer for about an hour.
Make the lemon glaze. Whisk powdered sugar, vanilla, and lemon juice until smooth but still somewhat thick. If it seems too chunky, add 1 tsp milk of choice until the desired texture is achieved. Remove the cake from the freezer and spoon lemon glaze evenly overtop. Set in the fridge for another hour. Slice into 12-16 bars and serve!