I received a very nice note from the neighbors. They thanked me for my apology and gently chastised me for the dog getting out. They explained that the cat in question was not her cat, but she was afraid for her cat since it was old and an outdoor cat. Then, they welcomed Dr. Man and I to the neighborhood and wished us well.
So, I think that went about as well as possible. We're going to dip into our emergency fund and get the privacy fence in place as soon as we can. I'm glad that it went civilly and they seem to have calmed down.
Dr. Man still votes with putting Dixie Dog poo on their yard, though. I've vetoed that idea.