In a medium saucepan, cover the tapioca with 5 cups of water and bring to a boil. Cover and cook over moderate heat until the tapioca is chewy and plump, 5 to 6 minutes. Drain and cool under running water. Transfer the tapioca to a medium bowl. Rinse out the saucepan.
Using melon ballers of different sizes, scoop out 2 cups of melon balls. Chop the remaining melon and transfer to a blender. Puree until smooth. Strain the puree, without pressing on the solids, to extract clear green juice. Add 2 tablespoons of the juice to the tapioca. Refrigerate the remaining juice.
In the same saucepan, bring 1/2 cup of the Moscato to a boil. Simmer until reduced to 2 tablespoons, about 10 minutes. Pour the syrup over the tapioca and refrigerate until chilled, about 15 minutes.
Spoon the tapioca and melon balls into Champagne flutes and top with the melon juice and chilled Moscato.