Modify your fence to go into the ground. Then warn the neighbor that if his dog comes into your yard you will be forced to shoot it for the safety of your children. If the threat of it doesn't work at least you will be on the record for warning him, trying to be polite, and being the bigger person, not to mention already contacting authorities who were unwilling to help. The next step you take if he continues to allow his dog to dig into your yard is to take him to court for trespass, and possible intentional infliction of emotional distress (although you wouldn't win on IIOED). If he fails to show up in court, and still allows his dog to dig into your yard - one day let it come through - a place something poison that the dog will eat in your yard. If the dog eats it go to your neighbor's house and tell him "look, I had this - insert item here - in my yard and you once again let your dog dig his way over, and now he is likley going to die if you don't get him attention soon." This way you will look like the good guy once again in that you will save him from his own irresponsibility. Sometimes neighbors are a pain in the ass, and sometimes only extreme measures will work, but believe me if it isn't the dog this person will just do something else.
And why should I have to go to these measures because of a freakin' dog? That is completely insane.
And, fwiw, I would completely agree with your statements in the case of annoyances, such as barking, etc. But the life of my family trumps all of the dog owner's rights in my book. The though of 5/8" of cedar boards seperating the 120 lb beast from a 3 year old girl is simply chilling. And I should have the right to let my daughter play in my yard without having the beast trying to break through the damn fence.
Anyway, luckily the lady who owned the house kicked him (the boyfriend) out before I had to do anything drastic.