Pure 2/3 cup basil leaves, 1 can coconut milk, and the curry paste in a blender. Cook mixture in a saucepan over medium-high heat, stirring, until bubbly, 2 to 3 minutes.
Add remaining can coconut milk, 1/2 cup water, the fish sauce, and sugar. Cook sauce at a low boil until thickened, 10 minutes.
Coat fish with 3 to 4 tbsp. sauce. Coat peppers with about 2 tbsp. sauce.
Oil cooking grate, using tongs and a wad of oiled paper towels. Grill peppers, covered, turning once, until browned, 4 to 5 minutes.
Transfer to a plate. Grill fish, covered, turning once, until just cooked through, 3 to 5 minutes. With a wide spatula, carefully transfer to another plate.
Spoon rice onto a large platter and top with fish. Thinly slice remaining basil. Spoon half of sauce over fish, top with peppers, and sprinkle with basil.