1. Make the strawberry syrup: In a pot, combine strawberry puree and sugar with ½ cup water. Bring to a boil for 5 minutes, then strain and set aside.
2. Make the batter: Preheat oven to 350 degrees. Spray three 9-inch cake pans with non-stick spray and line with parchment paper. In a mixing bowl, combine sugar, cake flour, and baking powder. In a separate bowl, whisk together eggs, egg whites, vanilla, oil, and milk. Stir wet ingredients into flour mixture until well-combined. Fold strawberry syrup into cake batter.
3. Lightly toss chopped strawberries in flour, then gently fold into cake batter. Portion batter into prepared pans. Bake for 10 minutes, then rotate and bake for an additional 10 minutes. Continue to bake until a cake tester comes out clean. Set cakes aside to cool.
4. Assemble the cake: While cake is cooling, make the cream cheese frosting. Using a stand mixer fitted with paddle attachment, mix cream cheese on low until smooth. Add butter and mix to combine until mixture is creamy and shiny. Add powdered sugar, salt, lemon juice, and food coloring (if using). Mix until fluffy.
5. On a cake stand, place one strawberry cake. Spread with ½ cup to 1 cup strawberry jam, then top with another cake. Repeat with remaining jam and cake. Ice completed cake with cream cheese frosting.