The two should stay separate. I am also a dog lover and I have had dogs and birds successfully, but they never were out to play together nor interacted together.
I do think you should acclimate your dog to the presence of the cage though. If your dog is barking frequently at the bird it can cause a lot of stress in your home. Your bird may not feel safe, which could complicate training and your bird-human relationship. Also your dog is clearly very upset about this adjustment.
I would do what you can to acclimate your dog to the bird's cage and surroundings to improve the household dynamic.. but not with the intent to have the bird and dog be friends. What has worked for me before when I had two dogs was to have two of everything. If I am giving attention and treats to one pet, give attention and treats to the other pet. Reward your dog when they are calm and silent around the cage. Don't reward the barking behaviour by giving your dog attention during. Currently when your dog barks at the cage you are shifting your attention to him/her.
It may help to move your bird to a travel cage and set him in your bedroom or separate location and use the main cage with your dog. Set the empty cage on the ground, let your dog sniff it, and reward your dog for being near the cage silently. Place the cage back in its location and try and work with your dog by approaching the cage and continue to reward the good behaviour. I do not know your dog so I can't know this will work, this is only a suggestion.
Your dog maybe overwhelmed by the change and breaking this down in increments may help. You need to enforce that the cage and bird in the cage are NOT threats. Your bird is NOT a toy. You can never let the dog and bird interacted directly together, but you do need a peaceful home. It is best for you, your bird, and your pup.