Combine the lemon juice and sugar in a small saucepan with 1/2 cup water. Bring to a boil, then reduce the heat to medium and simmer until the mixture is syrupy, 10 to 15 minutes. Transfer to a small bowl and set aside to cool.
Make the ice cream cake: Brush a 5-by-9-inch loaf pan with vegetable oil and line with plastic wrap. Scoop the lemon sherbet into the pan and spread into an even layer. Top with the vanilla ice cream in an even layer. Place a piece of plastic wrap directly on the ice cream and press with your hands to smooth the surface and firmly pack the ice cream. Freeze 30 minutes.
Uncover the ice cream and arrange the ladyfingers in a single layer on top, trimming them as needed to fit. Brush the ladyfingers with half of the prepared lemon syrup. Cover with plastic wrap and freeze until firm, at least 2 hours or overnight.
Make the meringue: Beat the egg whites and cream of tartar in a large bowl with a mixer on medium-high speed until foamy, about 2 minutes. Gradually beat in the sugar, then increase the mixer speed to high and beat until stiff shiny peaks form, about 4 more minutes.
Uncover the cake and invert onto a parchment-lined baking sheet; let the pan stand overturned until the cake slips out. Remove the plastic wrap and cover the cake completely with the meringue, using the back of a spoon to make swirly peaks. Freeze until firm, at least 2 more hours.
Preheat the oven to 500 degrees F. Bake the cake until the meringue peaks are golden, 3 to 5 minutes, or brown the meringue with a kitchen torch. Let the cake soften 10 minutes at room temperature before slicing. Serve with the remaining lemon syrup for drizzling.