If needed, slice a small portion of the domed tops off the cooled cakes, using a a sharp, non-serrated knife. Then split the two cakes so that you have four layers.
Place one of the cake layers on a cake decorating turn table or a cake plate, open crumb side facing up.
Spread half of the chilled strawberry compote over the cake, stopping 1/2 inch from the edge.
Place a second layer of cake on top of the strawberry filling. Spread a thin layer of the buttercream over the cake.
Place a third layer of cake on top, and spread with the remaining half of the strawberry compote.
Place the fourth layer of cake on top, open crumb side facing down. The layers should alternate as follows: cake, strawberry filling, cake, buttercream, cake, strawberry filling, cake.
Frost the cake all over with a thin "crumb coat" of buttercream, then chill the cake for 30 minutes.
Frost all over with a final coat of buttercream. You can smooth it out, but it doesn't need to look perfect.
Before the buttercream crusts over, press handfuls of the crumble mixture all over the cake.
If you have extra buttercream, pipe some swirls on top of the cake using a piping bag and a large open star tip (such as tip 1M). Decorate the top with freeze dried strawberries.