In a small bowl, whisk the ketchup, horseradish, and lemon juice. Season with kosher salt and a few dashes of hot sauce. Set aside.
Preheat the broiler. Meanwhile, on a large baking sheet, lightly butter the bread slices on both sides.
Fry the oysters: Line a baking sheet or platter with paper towels and set it by the stove. In a large, heavy-bottomed pot fitted with a deep-fry thermometer, add enough melted lard or oil to reach 4 inches up the sides of the pot. Begin preheating over medium-high heat.
While the oil is preheating, prepare the first batch of oysters: In a shallow bowl, whisk the corn flour and 2 tablespoons of salt. Work with 10 oysters at a time: Dredge them in the flour mixture, pressing it onto the oysters to cover, then shake gently to remove any excess. When the oil reaches 350°F, add the prepared oysters in batches, and cook, stirring occasionally with a spider strainer or slotted metal spoon, until golden brown, about 2 minutes. Transfer the oysters to the baking sheet as you continue cooking the remaining oysters.
While the last few batches of oysters are frying, broil the bread, turning once, until golden on both sides, about 1–5 minutes per side.
Spread 4 of the toast slices with a tablespoon of the prepared sauce. Spread each of the remaining 4 slices with a tablespoon of mayonnaise. Place 10 oysters on each of the sauce-topped toasts, then add a slice of tomato, a few leaves of lettuce, and the remaining toasts. Serve with pickle spears, lemon quarters, and hot sauce, if desired.