Five years is a long time to have never met the gal, I would go over introduce your self and be neighborly. Really, bake a cake, invite her over for coffee, and begin the process of knowing your neighbors truly. I know, she is a recluse, guess what? So am I ! But I made it an effort to speak with my neighbors when I moved in, told them I was there if they ever needed anything, and that I believed having good neighbors was importantly to me. Yep, during the holidays I even send cookies and treats to there doors. Down the road now 6 years my neighbors trust me, and at times I send eggs to them if I have extra's. My next door neighbor was a real poop, he and his wife were not happy when we moved in, and no matter what I did they would not be Friendly. I decided it was ok, my kids played, just as usual, and we were kind. One day he cut my hedge down almost to the ground ( yep he is right next door and he has a chain link fence and on the other side I have a beautiful lilac hedge. While at work he cut it down to about 3 feet to the ground, it was 8 feet tall! I talked with him, and I decided I was mad, but it would grow,it did. The next year he did the same thing, this time I had a registered letter from my land lord telling him he was not only destroying property, but that he was introuble. My neighbor was livid at me. But you know what, I smiled and told him that I was still here in case he needed me. Ok but in the mean time I prayed for him, and his wife. That same year, his wife went into the hospital , 5 days later he did to. Both died with in a week

this was 4 years ago, I have never had another next door neighbor as the daughter left it as it was and to this day it's still empty ( prayer lol ) neighbors can be a blessing or a curse, this man was a curse, but I was determined not to let him bother me, to this day I'm glad I never fought with him. I'm also glad the rest of my neighbors are cool with my birds and all is well. But my story is really , try to go to her, make the effort, try and try, and see if you can at least be on speaking terms and let her know you are there. Make her a nice cake, give her personally a dozen eggs, not him but her. Show her you care. You never know she may be quit nice. And with that you may be able to break through and she let you have the rooster with out strife.
I have always been taught, it's best to get along with the human kind, peace even if it means giving the rooster away. my neighbors are all arond me, I do not have the luxury of having lots of land, I'm blessed with a back yard and small front yard. But the key was to go to my neighbors and let them know me, this way if there ever was a problem, they and I have talked and this makes it harder for them to be angry over anything lol. The eggs make them happy to.