I can't offer a suggestion for Korean BBQ. For Korean food in general as others have mentioned Den Den has good food but the atmosphere is a bit of fast service restaurant and the chairs are a bit uncomfortable plus it is always packed.

Instead I would suggest Wok and Pot donwcity. It has good food, soju and other drinks and overall more food options. No BBQ Cooking at the tables but if you get the budae pot I think they do bring a little burner to the table and that is a great dish to share.