Sorry to hear you are having so much trouble and I am very sorry some poor chickies had to die in the process.
There are a few things to look at here from all perspectives:
*First it is right to make sure the damage is corrected, as best you can, and a reasonable sum is paid to cover the cost of the loss, both in actual bird cost and in the cost to get him back he was before the damage occurred.
*Secondly, every effort must be made to make sure this NEVER happens again. Because your dog was chipped by animal control if they are called out again the dog WILL be euthanized, period, and of course you don't want any more chickens to meet their maker either.
It is your responsibility to make sure that your dog is under control at all times and is contained within your yard - so fencing and leashing is vital and training and control is 100% your responsibility, so make sure that is taken care of.
It is your neighbors responsibility to protect his chickens, period. Fencing and predator control is 100% his responsibility and should be done to the best of his ability so a dog or anything should not be able to get to them in the first place.
On a soapbox note real quick: As an owner of Pit Bulls and Am Staffs I am concerned that your attitude is as such that you blow off aggressive behavior so cavalierly. I have trained my bullies to be gentle with everything from cats to kids and everything in between, including chickens - so it is completely possible to train them that way. Dogs are what we make them. Some dogs get very ingrained predator behavior so if that is the case - it is 100% your responsibility to make sure nothing like this happens again or your dog will be killed - by a pet owner or the city.
I say this not to talk down - just to be frank - I really wish you and your beloved pet much success and happiness.
Best of luck with it all.