Stir together 1 ¼ cups water, rice, ¾ cup coconut milk, 1 teaspoon oil, and ½ teaspoon salt in a medium saucepan. Bring to a boil, stirring often; reduce to a simmer. Cover and continue to simmer over low, undisturbed, until liquid is absorbed, 10 to 12 minutes. Remove from heat. Let stand, covered, for 10 minutes.
Meanwhile, heat remaining 2 teaspoons oil in a large saucepan over medium. Add curry paste, garlic, and ginger; cook, stirring constantly, until fragrant, about 1 minute. Whisk in remaining can of coconut milk. Bring to a boil over medium-high, whisking often. Stir in chicken, bell peppers, and remaining 1 ½ teaspoons salt.
Cover and cook over medium, stirring once or twice, until chicken is cooked through and peppers are tender, 6 to 8 minutes. Stir in mango; cook, uncovered, stirring often, until mango is heated through and sauce coats the back of a spoon, 2 to 4 minutes. Serve with coconut rice and lime wedges, topped with cilantro.
