I know this is an expense and unfair as this should not be your problem...but I would consider putting up a chain link fence with cemented poles along the way...along your fence line close or touching the old fence. If you have a handy man that knows how to do this, it would save some expense. I wouldn't count on your neighbor. Of course he is not worried, his dog is not threatened. Your babies are however, and unfortunately you get stuck with the problem.