Prepare an 8x3 inch springform pan by lining it with parchment paper that extends about 2 inches above the rim. This helps with easy removal. If using a regular cake pan, line it with plastic wrap and then parchment paper.

For the strawberry filling: Combine strawberry puree, 1/4 cup sugar, and 1 tbsp cornstarch in a small bowl. Microwave in 10-second intervals until thickened, about 30 seconds to 1 minute. Let cool.

For the key lime filling: Combine key lime juice, 1/4 cup sugar, and 1 tbsp cornstarch in a small bowl. Microwave in 10-second intervals until thickened, about 30 seconds to 1 minute. Let cool.

For the crust: Mix shortbread crumbs, 3 tbsp sugar, and melted butter. Press evenly into the bottom of the prepared cake pan and freeze until firm.

For the coconut ice cream layer: Beat 8 oz cream cheese and 1/2 cup sugar until smooth. Mix in 2 tbsp milk and coconut extract. In a separate bowl, whip 1.25 cups heavy whipping cream and 1/2 cup powdered sugar to stiff peaks. Gently fold about 1/3 of the whipped cream into the cream cheese mixture, then fold in the rest. Spread this mixture over the crust and return to the freezer.

For the strawberry ice cream layer: Beat 8 oz cream cheese and 1/4 cup sugar until smooth. Mix in 2 tbsp milk, strawberry extract (if using), and the cooled strawberry filling. In a separate bowl, whip 1 cup heavy whipping cream and 1/2 cup powdered sugar to stiff peaks. Gently fold about 1/3 of the whipped cream into the cream cheese mixture, then fold in the rest. Fold in the pink food coloring. Spread this mixture over the coconut layer and return to the freezer.

For the key lime ice cream layer: Beat 8 oz cream cheese and 1/4 cup sugar until smooth. Mix in 2 tbsp milk and the cooled key lime filling. In a separate bowl, whip 1 cup heavy whipping cream and 1/2 cup powdered sugar to stiff peaks. Gently fold about 1/3 of the whipped cream into the cream cheese mixture, then fold in the rest. Fold in the lime green food coloring. Spread this mixture over the strawberry layer and freeze until completely firm, 5-6 hours or overnight.

Once firm, carefully remove the cake from the springform pan. Smooth the sides with a warm offset spatula if needed and transfer to a serving plate. Decorate with Cool Whip, fresh strawberries, and lime slices. Let the cake sit at room temperature for at least 30 minutes before cutting and serving.