Generously season the pork with salt and pepper on all sides. Place in a slow cooker. Pour in the pickled carrots, dilly beans, and banana peppers along with their liquids. Top with the onion and pour the beer over top. Cook on high for 7 – 8 hours or until the pork is tender and pulls apart easily.
In a mixing bowl, combine the mayonnaise, sugar or honey, vinegar, ½ teaspoon each of salt and pepper, and whisk until creamy. Stir in the coleslaw mix, cover, and refrigerate until serving time.
When the pork is done, transfer the meat to a large dish or platter. Using a slotted spoon, remove the vegetables from the slow cooker and add to the pork. Using two forks, shred the meat and vegetables. Serve on gluten free hamburger buns with the coleslaw.