Whisk the mayonnaise, cayenne, 1/2 teaspoon salt and 1/2 teaspoon black pepper in a large bowl. Add the red peppers, white Cheddar and yellow Cheddar and gently fold until combined. Cover the bowl and refrigerate for at least 30 minutes before using.
Meanwhile, preheat the grill to high for direct grilling. Divide the ground chuck into four equal portions, 6 ounces each. Form each portion into a 3/4-inch burger and make a deep depression in the center with your thumb. Brush both sides of the burgers with the oil and sprinkle liberally with salt and pepper. Grill the burgers until golden brown and slightly charred on both sides and cooked to desired doneness, about 4 minutes each side for medium doneness. During the last minute of cooking, place a dollop of the cheese mixture on top of each burger and close the grill lid. Cook until the cheese has just melted, about 1 minute.
Place the burgers on the buns and top with the bacon, onions, tomatoes, pickles and lettuce if desired.