I love a good Moscow Mule.
The Red Stag has a Moscow Mule ( or did at one time), but I believe they use Jamaican ginger beer which contains pineapple juice so they get points taken off. I can’t remember if they use a copper mug though.
I’d highly recommend Haute Dish’s Minnesota Mule! They use a copper mug and (non-Jamaican) ginger beer.
Be forewarned, lately some places hold on to your credit card while you’re using the copper mug because too many rapscallions have stolen the copper mugs which aren’t cheap.
Any other places around town that offer the Moscow Mule?
I'd try Marvel Bar, Bradstreet, Cafe Maude, Eat Street Social and The Strip Club. All have incredible cocktail programs. I can guarantee they all offer a Moscow Mule, but I'm not sure about the copper mug. If you try any of these, please come back and post and let us know how it was!
Moscow on the Hill makes one and it was served in a copper mug as recently as mid-May, but I learned from a waiter recently that customers had been "keeping" the mugs so they're served in a regular glass now.