For the remoulade: Combine the mayonnaise, capers, chives, pickle juice, creole mustard, Old Bay, lemon zest and juice, and some salt and pepper in a small bowl. Whisk until well combined and smooth, about 1 minute. Cover and refrigerate.
For the crabcakes: Combine the eggs, lemon zest and juice, mayonnaise, Old Bay, salt and hot sauce in a bowl; whisk together until smooth and creamy, about 1 minute. Add the backfin crabmeat and the crushed crackers, and mix until well combined. Gently fold in the lump crabmeat. Divide the crabcake mixture into six equal portions; roll each into a loose ball and place in a large cast-iron pan.
Transfer the pan to the oven and bake until the crabcakes are golden brown and cooked through, about 15 minutes, flipping once halfway through. Arrange the crabcakes on a large serving platter. Top each with a dollop of the remoulade. Serve immediately.