I wonder if you could move that "temporary" structure a bit closer to your birds (next to it) and then pull one of your birds into the temporary enclosure, etc..... and maybe the neighbors will forget all about the hen? I was going to suggest secretly rehoming her and her "boyfriend" but word could get back. I'm not wild about recommending lying to the neighbors but they've had a long time to change their ways and some things never change. Like caring about animals or not.