In a small bowl, combine the shallot, red wine vinegar, black pepper, and kosher salt; let stand 10 minutes to mellow.
In another bowl, whisk the ketchup, prepared horseradish, lemon juice, Worcestershire sauce, and hot sauce until smooth; refrigerate until serving.
Spread the crushed ice over a large rimmed platter and chill the platter.
Working one at a time, scrub the live oysters under cold running water, then, using a folded towel and an oyster knife, insert at the hinge and twist to pop; slide the blade under the top shell to release, discard the top shell, and run the blade under the meat to free it from the bottom while keeping the shell level to retain the liquor; check for shell fragments and turn the meat cup-side up.
Nestle each shucked shell into the ice and serve immediately with the mignonette, cocktail sauce, and lemon wedges.