In a small bowl, mash 1 cup berries with the lemon juice and 1/3 cup sugar; stir in the tapioca and let stand for 15 minutes.
In a heavy, medium saucepan, melt the butter over medium heat. Add the berry mixture and remaining 1/3 cup sugar and cook, stirring often, until hot, about 3 minutes. Add the remaining berries and bring to a boil, stirring often, for 3 minutes; lower the heat, cover and simmer, stirring often, until the tapioca is tender, about 15 minutes. Let cool.