I too have 2 dogs who share the house with the birds. It can work very well, but you do have to be extremely cautious and never let the puppy alone with the bird for even a second, both for the safety of the bird but also the puppy!
I grew-up with a CAG, we got him when I was 9 years-old and we had no dog at the time...We added a Pug puppy, Rudy, when I was 15 years-old and Jagger, our CAG, was 5-6 years-old, and being a tiny little Pug puppy, Jagger was dangerous to her, not the other way around! He was immediately jealous of Rudy and would dive-bomb her...He could have easily grabbed her with his feet, as Rudy fit into the palm of my hand when we first brought her home...So we had to always make sure that we were holding Rudy securely and we never let them out together for the first few months, until Jagger started to calm-down and realize that he wasn't going to lose any attention or time, and that Rudy was not a danger to him...
Now flash-forward and I had dogs in my adult life BEFORE I had any birds (except for breeding-pairs, but they weren't out and about in the house)...When I added my Lita, my Quaker Parrot, as a little baby to the house that was already "owned" by an Australian Cattle Dog and a male Shar Pei, it was completely different. The dogs were absolutely horrified by Lita flying around, they were scared to death of her, and of the rest of the birds that followed...and my birds have no fear and will go after the dogs if they want to, especially Bowie, the Green Cheek, he will walk right over to Lola, my Cattle Dog, and just take a huge bite out of her if he feels like it...
***Here's my best advice for you, regardless of what breed of dog you decide on...In my experience with multiple birds and dogs and having the birds come first and then the dogs come first, etc., regardless of the situation, It's much easier to train the puppy/dog to stay away from the bird and to never go after the bird than it is to try to train the parrot to stay away from the dog and not go after the dog/puppy!!! This is why I would suggest ALWAYS getting a young puppy so you can easily train it to not chase/go after/attack the puppy....With parrots, they are naturally prey-animals, so it's pretty difficult to train a parrot to not think of a dog as a predator. In my experience they can get along well, but it's just easier if you start-off with a puppy and you train the puppy to respect the bird and that the bird is not a toy or something to chase...And then eventually if the puppy is well-trained to respect your CAG, then your CAG will gradually become more and more comfortable with the puppy...And that's important, because a CAG could easily kill a small puppy, and could take out an eye of a large dog, as well as lay some vicious bites on them...
***I don't know what breed of dog you're thinking about, but the only breeds I would absolutely stay away from are the "hunting" and "bird" dogs...So no hounds, like Beagles, Bassets, etc., and definitely no "bird" dogs, such as Pointers or Spaniels, because they all possess natural instincts to just innately go after birds and small animals. I have an Australian Cattle Dog, I've actually had two of them, both lived with birds including my mom's CAG, and they are fine with the birds (but they are also extremely intelligent and very easy to train; they also aren't "hunting" dogs, just herding dogs)...But my mom and stepfather have a French Brittany Spaniel that my stepfather got as a young puppy and he trained her to hunt game birds, and she does not mix well with birds. It's an interesting dynamic, but not one I'd want to have to deal with every day.
