Preheat the oven to 350 degrees F. Spray a 9-by-13-inch baking pan with nonstick cooking spray, then line with parchment paper and spray the paper.
Stir to combine the graham cracker crumbs, melted butter and 1/4 cup sugar in a large bowl. Press evenly into the prepared pan and bake for 10 minutes, then set aside to cool. Reduce the oven temperature to 325 degrees F.
Beat the cream cheese and remaining 1 1/3 cups sugar in a stand mixer fitted with the paddle attachment on medium-high speed until light and fluffy, about 3 minutes. Turn the mixer to medium-low speed and add the eggs one at a time, making sure not to overbeat the batter.
Reduce the speed to low and add the rum, flour and triple sec until just incorporated. Remove the bowl from the mixer and fold in the strawberry puree, making sure to swirl the strawberries throughout. Pour the batter into the prepared pan and bake until the cheesecake is just set but still slightly jiggly in the center, 38 to 42 minutes.
Let the cheesecake cool for 1 hour at room temperature, then transfer to the refrigerator and chill for at least 4 hours and up to overnight.
Cut into bars and top with whipped cream, sliced strawberries and mint sprigs.