Prepare a charcoal or gas grill for medium heat (350° to 450°; you can hold your hand 5 in. above cooking grate only 5 to 7 seconds).
Meanwhile, in a large bowl, gently mix chicken, cheeses, garlic, salt, and pepper. Form into 4 burgers about 3/4 in. thick, making a slight depression in center of each patty (this helps keep burgers flat as they cook). Put on a plate, cover, and refrigerate until ready to grill.
In a blender, whirl mayonnaise, basil, lemon juice, and lemon zest together and refrigerate until ready to use. With a silicone brush or oiled paper towels, lightly oil cooking grate.
Grill burgers, covered, turning once, until browned and no longer pink inside (cut to test), 8 to 10 minutes total. Transfer burgers to a clean plate and place buns, cut sides down, on grill until lightly browned, 30 seconds.
Spread toasted sides of buns with basil mayonnaise. Set a chicken patty on each bun bottom and top each with 2 strips bacon, a tomato slice, and a lettuce leaf. Cover with bun top.
Note: Nutritional analysis is per burger.