I think that @leopardsponge wanted the neighbour to complain to her first, because she (the OP) was convinced that had the neighbpur done so, she would have just needed to explain to the rightly annoyed neighbour, that her other dog doesn't bark (are they supposed to be grateful about that?), and this one is just being a dog, so that is that. Maybe she then thought that the neighbour would think,
"Oh gosh yes, why hadn't I realised that the poor dog is just being a dog. I must stop being so selfish about missing my last bit of sleep every single night."
Unfortunately for the OP, she did not get the result she was hoping for, actually was probably expecting, and now doesn't know what to do. I doubt very much that she likes the idea of getting up at 6.50 in the morning to take her dog for a walk, but that is what she needs to do if he can't wait any longer! I am wondering if because it hadn't seemed to even cross her mind to take her dog for a walk at 7.00am, if she is one of these people who get a dog, and never take it for a walk.
OP, if you do happen to still be reading these comments, your dog needs at least two good walks a day, and depending on your breed of dog and it's age, they need to be for at least half an hour a day, to up to about 5 hours a day if you have a husky or similar. Your dog desperately needs walks for mental stimulation, as well as exercise - obviously your other dog needs these things too!
You can get away with just taking your dogs out for a quick walk of 5 to 10 minutes first thing in the morning, if you then take them out for two proper walks later. If you have young children in the house who you obviously can't leave at home if their isn't another adult in the house, then put a lead on the noisy dog and take them into the garden, but at the first hint of a bark, take him straight back in again.
I do have a suggestion for the OP's neighbours though, I wonder if the OP's nearest neighbours would be willing to be woken up at about 4 or 5 am for a few mornings, and could borrow a cockeral from a friendly farmer...?