Start with 12-inch cakes. If necessary, trim top of each with long serrated knife to make it level, then cut
horizontally in half. Put each layer cut side up on an 11-inch cardboard round. Brush tops generously with syrup. Stir jam until smooth, then spread about 2/3 cup on one layer. Invert another layer (on cardboard) onto jam. Discard top cardboard round and spread about 2 1/2 cups frosting on top of layer. Scatter a layer of blackberries over to cover frosting. (If berries are 1 inch or larger, first halve them lengthwise.) Slide the third 12-inch layer, syrup side up, onto berries (discard cardboard) and press down gently. Spread about 2/3 cup jam on layer and invert last 12-inch layer (on cardboard) onto jam, discard cardboard.