I say get the hens. My neighbor's dog sits and whines at our fence half the day, and has even made attempts at digging under, but the neighbor does keep her on a tether.
Ultimately, you are not doing anything illegal, and really, if someone has a "nuisance animal" in this situation, it's her. The citizen service bureau (or whatever equivalent you have) would be more likely to get on her back for neighbor complaints than yours.
But because of the ill-will, you really will need to watch your back and make sure you really are sticking hard and fast to all the legalities. She will look for the tiniest thing to get you for.
Hens have made my life SO much better. A greenhouse would be cool, (I really do want one...) but not nearly as loveable as hens.
Just my 2 cents - (oh, and i have a rooster. All my neighbors are lovely about it. They even enjoy his crowing! But I'm lucky, I guess.)