Hello, celiac disease sufferer here! I absolutely love Korean food, but it can be difficult to order gluten free due to the ubiquitous presence of soy sauce. This is why I tend to stick to grilled meats and choose the unmarinated ones. Palace has a great selection of unseasoned meat and you grill it on a clean metal surface at your table, which makes for pretty low cross-contamination risk! I do not know if they have gluten free soy sauce. I didn’t ask as I often bring it along to Japanese and Korean restaurants and add it to my own dish (just put the bottle away if you do this as outside food and drink are generally frowned-upon). Palace does provide sesame oil with salt and pepper for dipping, which is a delicious gluten free condiment for the meat.
The only thing I missed from the grilling experience was the lettuce, raw garlic and jalapeño, and pickled radish that sometimes accompanies Korean BBQ at other restaurants. Just a preference, but I missed these flavors and textures.