Make the burger sauce. Stir together the mayonnaise, bbq sauce, ketchup, mustard, and paprika in a small bowl. Refrigerate until ready to serve or up to 5 days.
Make the burgers. Gently combine the ground round, mayonnaise, Worcestershire sauce, garlic salt, and pepper in a large bowl. Shape the mixture into 6 (¾-inch-thick) beef patties.
Preheat the grill to medium-high heat. Grill the patties for 3 to 4 minutes on each side, or until almost done. Place the cheese slices on the patties and let stand until melted.
While the burgers cook, grill the red onion slices alongside for 2 to 3 minutes on each side, until tender.
To serve, spread the burger sauce on the bun bottoms. Top with the patties, bacon, pickles, tomato, red onion, and lettuce. Replace the bun tops and serve immediately.