For an adult dog, I would definitely recommend a good rescue. Also be very clear that you have a bird and to ask how any potential candidates are around cats, birds, squirrels when on walks. In my experience a dog reactive on a walk is going to be more likely to be interested in that same creature in the home. A lot of rescues also have a foster to adopt system where you can do a 'trial run' and see if the dog meshes with your household. I personally would also recommend a smaller dog, any dog can do damage, but a Chihuahua is simply not doing to do as much, as quickly as say a Shepherd or a Dane.
As far as breed specifics go I would lean toward the Toy group or any dog bred for companionship, Shih Tzu's come to mind as they are typically pretty chill dogs, although every dog is an individual. For example we had a Cavalier King Charles come to daycare that was very drivey when the breed is known for being a royal lapdog (royal in every sense of the word lol).
Hypoallergenic is also just a buzzword used to sell puppies to people with allergies. No dog is hypoallergenic, low shedding breeds emit less allergy inducing dander as the fur doesn't readily shed and release the dander; but it still exists. Beware of anywhere selling "hypoallergenic, tiny, micro, teacup, teddy bear" nonsense as it's all just buzzwords to get your attention. I would also avoid Doodles and the like at all costs due to poor breeding practices.
Training with any dog though is going to be your best friend, along with constant supervision.
I appear to be in the minority in regards to birds with dogs as my current lovey and my past bird have always been allowed to be out and around the dogs. Sandi was introduced to Valkyrie one on one when we brought her home and began the association that the inside bird is not the same as the outside birds she's allowed to chase. She needed to learn about the new addition and Valkyrie needed to learn and get used to her as well. I do trust Sandi around Valkyrie (even if Valkyrie plops to the floor) however I would never leave them alone together without supervision, as the saying goes "when the cats away, the mice will play," and we are the cats keeping our mice in line, lol. As for Angel, she is allowed to watch Valkyrie from the hallway, but she also has little regard in her senior years.