Prepare the cake mix as the label directs for a 9-by-13-inch cake. Bake and let cool 15 minutes in the pan on a rack, then transfer the cake to the rack to cool completely.
Trim the domed top of the cake with a long serrated knife to make it level. Cut the cake in half horizontally to make 2 layers. Trim a 3/4-inch strip from each long side; reserve the strips and trimmings.
Cut the carton off the ice cream and set the ice cream block on a cutting board. Slice lengthwise into thirds.
Arrange the ice cream on the bottom layer of the cake. Let soften, then smooth with an offset spatula to cover the cake, leaving a 3/4-inch border on the long sides.
Press 1 reserved strip of cake against each long side of the ice cream layer. Reserve the remaining 2 strips of cake. Place the second cake layer on top of the ice cream. Freeze 1 hour.
Cut off a corner of the frozen cake to look like a bite. Remove the ice cream from the trimmed bite and reserve the cake trimmings. Return the cake to the freezer.
Preheat the oven to 350 degrees F. Crumble the reserved cake strips and trimmings on a baking sheet. Bake, tossing occasionally, until golden, 10 minutes. Let cool, then crumble together with the freeze-dried strawberries.
Finely crush 1 tablespoon of the strawberry-crumb mixture; combine with 1 tablespoon frosting to make it pink. Cover the whole cake with the remaining white frosting. Press the strawberry-crumb mixture into the frosting, without covering the bite or the opposite short side.
Spread the pink frosting inside the bite. Add a line of crumb mixture at the bottom of the bite. Freeze until firm, at least 15 minutes. Insert a wooden spatula handle or large craft stick into the cake before serving.