I just had the very same problem and actually lost some birds. Luckily my neighbor felt awful about it and keeps his dog away now. But what you need to know is this: if that dog is on your property and threatening your birds, it is no longer a pet, it is a predator. You can shoot it, kill it, trap it, whatever, if it is on your property threatening your livestock (chickens count as livestock). So maybe you're not comfortable killing it. I know I wasn't, so my plan was, if the neighbor did not cooperate, to catch the dog the next time it was on my property and find it a new home far away, whether that be by dropping it at a shelter or placing it myself. I was going to inform my neighbor, if he was uncooperative, that it was fine if he didn't want to pen or tie his dog, but if the dog was on my property again, it was going to disappear. 
Luckily it did not come to that, but if it had, I would have told him this and 100% followed up on it. I think that since the owner, in your case, is unwilling to work with you, maybe you need to inform them of your rights to protect your birds, and tell them that if the dog is on you property again it is going away permanently. That, they may take seriously. 
And if they don't listen, you need to follow through with it and make it disappear in whatever way you're comfortable with.  Believe me, right now he may just be chasing, but eventually she'll catch and kill one and more.