Place the milk and 1 cup of the cereal in a medium bowl and let sit until the milk takes on the cereal flavor, about 30 minutes. Strain the milk and discard the cereal. Set the milk aside.
Meanwhile, melt the butter in a medium saucepan over medium-low heat. Add the marshmallows and stir continuously until melted, about 5 minutes. Remove from the heat and stir in the remaining 3 cups cereal until coated.
Press the mixture into a 9-inch springform pan and let cool completely at room temperature, about 30 minutes.
Combine the gelatin with 1/4 cup water in a small microwave-safe bowl and set aside to soften, about 5 minutes. Beat the cream cheese, confectioners' sugar, sour cream, lemon juice and vanilla together in a large bowl with an electric mixer on medium-high speed until completely smooth, about 1 minute. Scrape down the sides of the bowl and add the cereal milk. Continue to beat on low until completely incorporated, about 1 minute.
Microwave the gelatin, stirring as needed, until melted, about 30 seconds. Add the gelatin to the cream cheese mixture and beat on medium speed to combine, about 30 seconds. Pour the cream cheese mixture on top of the cereal crust and smooth out the top if needed. Refrigerate until completely set, at least 4 hours and up to overnight.
Evenly divide the white chocolate among 3 small bowls. Heat the heavy cream in a small saucepan until boiling. Remove 2 tablespoons of the cream to a small bowl and color with the yellow food coloring, then pour over the chocolate in one of the bowls. Color another 2 tablespoons of the cream with the blue food coloring in the same manner and pour over the chocolate in another bowl. Color the remaining 2 tablespoons cream with pink food coloring and pour over the last bowl of chocolate. Let all three bowls of chocolate sit for 5 minutes, then stir until completely smooth.
Open the clasp of the springform pan to unmold the cheesecake. Drizzle the colored chocolate in alternating sections around the edge. Sprinkle more cereal on top of the cake in a ring inside the colored-chocolate edging. Pipe or dollop whipped cream inside the cereal circle and top each dollop with a Lucky Charms® marshmallow.
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