So, I figured out why the joke doesn't work for me. Because Balin mentions the Green Dragon immediately before, it makes the joke itself rather confusing. Since Balin is the one who brought up the punch line, it feels kind of like a "Dad Joke" to me.About the naming of the inn: I like it, but I'm not sure that Bard is the one to whom Balin should have made the suggestion. Would not the proprietor be more appropriate? Or is that Bard?
Edit: Also, Balin says that he was inspired by the Green Dragon in the Shire, after which, Bilbo corrects him that it was, in fact, the Green Dragon. I'm thinking that your brain may have just revolted against the error while typing it out.
Re-Edit: Ok, I see what you're doing now. I'm not really sold on the joke, but I retract my earlier objection.