Mix the ketchup, the juice of 1 lemon, 2 tablespoons vodka, the horseradish and hot sauce in a medium bowl; set aside.
Preheat the broiler. Whisk the olive brine, olive oil, thyme, the juice of the remaining lemon and the remaining 2 tablespoons vodka in a medium bowl. Add the shrimp and toss to coat. Wrap each shrimp with 1/2 slice bacon and arrange on a baking sheet; pour the juices from the bowl over the shrimp. Broil until the shrimp is cooked through and the bacon is crisp, about 4 minutes per side. Let cool slightly.
Skewer each shrimp with a cocktail pick, placing an olive inside the curve of the shrimp. Add a cube of lettuce to each skewer. Serve with the prepared cocktail sauce.