Heat a medium skillet over medium heat. Add the ground beef, green pepper, and onion. Cook while breaking up the beef into chunks with a spoon or spatula until browned. Drain off any excess liquids. Stir in garlic powder, mustard, ketchup, and brown sugar. Mix well. Reduce heat and simmer for 30 minutes. Season with salt and pepper.
Cut the dinner rolls in half and spread butter on the cut side of each bun. Place cut side down into a skillet over medium heat and toast until golden brown on the bottom.
Pour queso dip into a microwave safe dish and microwave until warm. Place some of the sloppy joe meat onto the top of the bottom half of each slider bun. Top with queso dip and banana pepper rings. Place the top half of the bun onto the slider.