Preheat oven to 350 degrees F. Line a 9x13-inch baking pan with parchment paper.
Prepare brownie mix according to the package directions for cake-like brownies. Stir in the chocolate chips and pour the batter, minus one cup, into the pan. Set the reserved one cup aside for later.
Layer the ladyfingers upside down in a single layer on top of the brownie batter in the pan and gently push down. Combine the coffee and rum in a small bowl. Use a pastry brush to brush the ladyfingers until soaked.
In a separate bowl, beat together the mascarpone cheese, cream cheese, sugar, egg whites and vanilla until smooth. Transfer to the pan and carefully spread over the ladyfingers until covered. Drop spoonfuls of the 1-cup reserved brownie batter onto the cream cheese layer and use a knife or toothpick to create marble swirls.
Bake for 40 to 50 minutes, or until the center is almost set and the brownies begin to pull away from the sides of the pan. Allow to completely cool in the pan before slicing and serving.