It CAN be done. But it's not easy. My dogs (and even the cats) know that prey animals are basically furniture. They are not allowed to chase them, jump or sleep on their cage, stare at them, or antagonize them in ANY way, because I don't think prey animals should have to live in fear. With that said, I would NEVER trust a dog to be loose with a bird unsupervised, even for 1 second, and mostly not supervised, depending on the training level and age of the dog (younger dogs are more impulsive, of course). I control my dogs in my house for a long time (either a leash, x-pen, or kennel). They aren't just "loose" to act like fools at any time, until they understand that the house is not a gymnasium.
But I have malinois. They're nuts (and side note, EXCEPTIONALLY high drive). But, I have malinois BECAUSE I compete with them, so they are at varying levels of training, depending on their age and experience, but the goal is HIGHLY trained. If I can call the dog off a helper, I can call her off a bird, lol. Nothing will EVER be as drive inducing as the helper. But, the goal is that should never even come up. Either the dog is trained enough that it is okay, under complete supervision, and consistent control (for instance, I'm expecting the dog to DO something, even if that something is just stationing), or it's not trained enough to be trusted loose with a bird at all. I also have the advantage that my dogs can spend a good deal of time outside, even though they are "indoor" dogs. I live in the country, and the dogs have the ability to spend much of the time that one of us is home and awake, outside, which gives me more freedom with the birds.
I agree that I'm not sure I like the puck. I like a LOT of clarity in my training, because I've found that my animals prefer it, and it is far more effective, and I just don't think the puck will do it. While I would say I train my dogs roughly 90% positively, I have no issue with someone using an e-collar for this, BECAUSE the results could be so catastrophic if it is not VERY clear that there are rules that are not optional. But I would use an e-collar if that is the route you choose to go, not the puck. I'd want absolute control over the timing and intensity of the correction. As an aside, a good e-collar ranges from not being able to feel it at all, to very high. The working level is where the dog can JUST start to feel it, I've tried it on myself, and it felt like a bug crawling on me (and actually, all of my dogs find the vibration to be far more punishing). Your dogs would be far less likely to be afraid of that. You could also train this positively. Basically, teach the dogs what you DO want, and reward that (stationing would be a great example). Absolutely NEVER reward ANY sort of attention on the bird.
