Prepare a grill for cooking over medium heat.
Combine the mayonnaise and ketchup in a small bowl and set aside.
Heat a medium skillet over medium heat; add the bacon and cook, turning once, until crisp, 7 to 8 minutes. Transfer to a plate covered with a paper towel and drain; chop.
Gently combine the beef, garlic powder and 1/4 teaspoon each salt and pepper in a bowl. Divide the mixture into 8 portions and form each into a 3 1/2-inch-diameter patty; put 4 patties on a work surface. Toss the bacon with the cheese and spoon a quarter of the mixture on top of each of the 4 patties, keeping the edges clear. Place the remaining patties over the cheese and pinch the perimeter to seal the filling in.
Grill the patties with the lid closed, turning once, 9 to 10 minutes for medium doneness. To assemble, put the bottom half of each roll on a plate and top with a lettuce leaf, tomato slice and patty. Spoon the mayonnaise mixture over the patties, then top with the onion slices and the remaining roll halves. Serve immediately.