It should be fine. Hold the rope and take it with you.
You can bring dogs. We just went there yesterday and many people brought dogs.
Now you have to park at the mountain gate and take the scenic shuttle bus into the mountain. Generally, the staff at the gate will prevent you from taking your dog into the mountain.
Last year, I also saw people holding dogs, but for safety reasons, it is recommended to keep the leash on.
If it is not a violent dog, you can put the dog on a leash and negotiate with the staff.