In a large skillet, melt butter over medium heat. Add onion, and cook until translucent, 4 to 6 minutes. Add bell peppers, and cook until soft, 6 to 8 minutes. Add garlic, and cook until fragrant, about 1 minute. Remove from heat, and reserve.
In a large bowl, add lemon juice, parsley, mayonnaise, mustard, salt, chives, red pepper, black pepper, and eggs. Whisk to combine. Add reserved onion mixture, crawfish, bread crumbs, cornmeal, and melted butter. Gently combine with a rubber spatula. Using a 1 oz. spring-loaded scoop, form crawfish mixture into balls, then slightly flatten.
In the bowl of a food processor, combine red pepper, sour cream, basil, and olive oil. Pulse until combined and transfer to a small bowl. Set aside.
In a large skillet, heat 3 Tbsp. oil over medium to medium-high heat. Working in batches, cook cakes 3 to 4 minutes, until golden brown and beginning to crisp. Using a fish spatula, flip cakes, and cook 2 to 3 minutes more. Remove from heat, and let cool on a wire rack. Serve with reserved red pepper remoulade.