Fill a large bowl with ice and water; set aside. Place the orange juice, sugar, and ginger in a small saucepan over medium heat; bring to a boil. Let simmer, stirring occasionally, until sugar has dissolved and syrup has thickened, about 15 minutes. Remove from heat; strain syrup into a clean bowl; add the Cointreau; stir to combine. Set bowl in ice bath, or chill in refrigerator, until syrup is cold.
Using different sizes of melon ballers, cut balls from the honeydew and canary melons. Place balls in a medium bowl; add 1/2 cup cold syrup and orange zest. Toss to combine.
To serve, arrange cantaloupe on a platter. Spoon melon balls on top. Serve extra melon balls on the side. Drizzle melons with syrup; garnish with mint. Serve with Orange Lace Cookies and remaining syrup on the side.