In shallow container, whisk together juice of 1/2 lime, soy sauce, fish sauce, sesame oil, garlic, ginger, and honey. Add pork to container, cover, and let marinade in the refrigerator for 4 hours or up to 12 hours, flipping pork once.
In separate container, whisk sugar and salt with 3 Tbsp hot water until sugar is dissolved. Stir in rice vinegar. Add carrots, cucumber, and radish; toss to coat and let sit for at least 30 minutes.
In small bowl, stir together mayonnaise and gochujang or sriracha.
Build medium-hot fire in charcoal grill, or heat gas grill to medium. Grease grill grates. Place pork tenderloin on grill, and cook, flipping only once, until internal temperature reaches 145 F, 5 to 7 minutes per side. Let pork rest for 10 minutes and then thinly slice.
Brush both sides of flatbreads with oil. Place flatbreads on grill grate and heat until crispy and darkened on bottom, about 2 minutes. Be careful not to burn bread. Remove flatbreads from grill and top heated sides with pork and cheese. Return flatbreads to grill and heat until cheese is melted and undersides of flatbreads are darkened and crispy. Remove flatbreads from grill and top with pickled vegetables, green onion, and cilantro. Drizzle on chili mayo and then squeeze remaining lime juice overtop.