Whisk together blackberry jam, balsamic vinegar, and olive oil in a small bowl. Season with salt and pepper and set aside.
Using a melon baller, scoop out flesh of cantaloupe and honeydew. Slice radishes into ⅛-inch thick rounds.
Toss together melon balls, radishes, mozzarella, mint, and blackberries. Arrange on a serving plate, then drizzle with olive oil and sprinkle with coarse salt. Serve alongside blackberry balsamic vinaigrette.