Well you say you've never really spoken to the wife. Speaking as a neighbor who has felt isolated from most of my neighbors, this makes it really, REALLY easy to not appreciate something that they do/something of thiers. She may not like the rooster by the simple fact that she doesn't know you and it brings in an unfamilier sound that she feels she shouldn't have to put up with. Being familier, even a little familier, with your neighbors and knowing that they care about you can make a huge difference.
What I suggest is that you go over and talk to the wife. Tell her that you know the rooster bothers her, and that you wish to resolve the problem without getting rid of him, because you and your family love him so much. Don't bring in the fact that you may legally keep him, as this could cause her to go on defense. When do you let your rooster out of the coop? If it's earlier than when she gets up in the morning perhaps that's why the rooster bothers her. I know I get grouchy if I don't have enough sleep, and with 5 children she could be quite stressed already.
This - I think going over and showing concern for the wife's feelings is a good idea. If it is the fact that he's crowing at 5:30 and waking her up, that can be remedied.