I agree with you about not wearing earplugs at night with children in the house. Even now with my children older, I still need to hear what is going on at night (preferably not noisy neighbours or dogs or motorcycles rumbling down the street). I can't sleep with earplugs.
Kusanar had good suggestions for approaching the neighbour. You could also take the recordings you have with you if they are skeptical about the noise. Do you have any recording of your work meetings with the chickens making noise in the background. That would be helpful to show them how it is affecting not only your home life, but work as well. Good luck. I hope it works out well for you and the neighbour.