Prepare the Shrimp Broth: Peel and devein raw shrimp, reserving shells. Cover and refrigerate shrimp until ready to use. Rinse shells with cold water until clean. Place shrimp shells, 5 cups water, chopped carrot, celery, onion, butter, kosher salt, and bay leaf in a large stockpot. Bring to a boil over high, and boil, stirring occasionally, about 5 minutes. Reduce heat to medium-low, and simmer, covered with lid slightly ajar, until reduced to about 3 cups, about 45 minutes; skim off and discard any foam or other impurities that rise to the top. Pour broth through a fine mesh strainer into a bowl; discard solids.

Prepare the Okra Soup: Heat a large gumbo pot or 5- to 6-quart Dutch oven over medium-high. Add olive oil, garlic, onion, bell pepper, 1 tablespoon of the butter, and 1 teaspoon of the salt. Cook, stirring often, until vegetables soften, 4 to 6 minutes. Stir in corn, lima beans, and plum tomatoes until well combined. Stir in canned diced tomatoes, reserved Shrimp Broth, and 2 teaspoons of the salt; bring to a boil. Reduce heat to medium-low, and simmer until mixture is slightly thickened, about 20 minutes.

Meanwhile, melt remaining 1 tablespoon butter in a medium skillet over medium-high heat. Add the reserved shrimp, and season with pepper and 1/2 teaspoon of the salt. Cook, stirring occasionally, until the shrimp begin to turn pink, about 2 minutes. Add sliced okra, ground ginger, and fresh lemon juice. Continue cooking, stirring occasionally, until okra and shrimp are cooked through, about 2 minutes. Remove from heat.

After soup has simmered 20 minutes, add shrimp-and-okra mixture. Cook until shrimp are heated through, 1 to 2 minutes. Season with remaining 1/2 teaspoon salt, and stir. Ladle into individual bowls. (If desired, spoon rice evenly into bowls before adding soup.)