Maybe you could call The Dept. of Agriculture to see if anyone can answer the question for you as to whether or not the tree will be damaged by what you plan to do. Sometimes when you cut a tree you have to spray it with some special black stuff(?) Not sure what it is, but I'm sure they, or Home Depot, can tell you. Guess it keeps the bugs out and seals up the cut area.
If a tree specialist tells you it will be fine, and how to do it correctly, then you could take that information with you to ask the neighbor's permission along with the person's name who you spoke to on the phone. You can try being really polite, explain what happened to your fence and then you can hope the neighbor is nice!