Combine first 4 ingredients in a small bowl; brush both sides of mushroom caps with sauce mixture.
Place bell pepper on grill rack coated with cooking spray; grill for 10 minutes or until blackened. Place in a zip-top plastic bag; seal. Let stand 10 minutes. Peel. Place mushroom caps, top sides down, and onion on grill rack coated with cooking spray; grill 4 minutes on each side or until tender. Place 2 cheese slices on top of each mushroom cap; cover and grill 1 minute or until cheese melts.
Spread 1 tablespoon ketchup and 1 1/2 teaspoons mustard over top half of each bun. Place mushroom caps on bottom halves of hamburger buns. Separate onion into rings; arrange onion slices evenly over mushrooms. Top each serving with 1 lettuce leaf, 2 pickles, and top half of bun.