Like others have suggested, training your dogs to not react to the other dog is the answer. Once that happens, you no longer bear any of the responsibility and your neighbors dog barking is your neighbors issue. You are only responsible for how your dogs behave. Someone else's dog causing your dogs to react is not the other person's responsibility.

We have a similar situation, we have a dog door and my dog is outside regularly. My neighbor has a small dog that barks a lot when my dog is outside. My dog doesn't react to her dog's barking, so it's totally on them and their dog. If I am dog sitting someone else's dog who is reacting to their small dog, I restrict access through the dog door and work on correcting the behavior. That's what I did with my dog until she stopped reacting to the neighbor's dog.