Have seen dogs on a leash numerous times at Lookout Mountain GC. I have no idea what the club rule is, as I do not have a dog.
On the other hand, dogs often roam the course. So if you had a good whistle or call, or a dog treat, you could probably have a friend for a few holes.
At my home course, dogs used to be allowed. In fact, the golf pro emeritus loves dogs and always had his small poodle at the shop. His poodle had a special friendship with another member, and would always accompany that member, and ride in his cart as he played. The poodle would fall behind at times, and run towards the golf cart. Jack would slow the cart down, and 'Bogey' would speed up and jump in the cart. She had a fondness for cheese crackers and Jack always had that treat. After finishing play, 'Bogey' returned to the shop.
There is still a small marker labeled 'Bogey' in the small flower plot next to the golf shop, still there after about 10 years now.