We had a similar problem not too long ago, but we were on the other side of it. Our neighbors built their house 10 feet from the tree line at the back of our property. A big storm came through and knocked a branch down on their yard. They tried to convince the insurance company that the tree was dead and it should affect our rate. BUT the tree wasn't dead therefore it classified under the "act of God" policy and they were required to take care of it.
Long story short, it was on them. Had the tree been dead, we would have been the ones responsible for clean up. Most likely they don't want their insurance rate to raise up. Keep on them to get it fixed. You can prove that the tree was dead, they failed to take care of it and it caused damage to the fence.
I hope they'll be willing to work with you, I'd hate for you have to escalate. Has your insurance contacted their insurance? Make them fight it out, that way you don't have to get in a fight with your neighbor.
Good luck!