Hi! I read the whole thread to try to be useful and not repeat.
A few things to consider:
A 5 chicken flock means they might be personally attached to the hens- makes it harder to get rid of some.
But it also means they might be very aware of who is noisy and who is quiet. It's possible they are more attached to the quiet ones and wouldn't mind rehoming the noisy ones.
Also, if it's been 3 years and the same hens, some may have stopped laying. Or will slow down soon. It mostly depends on the breeds they have. I've read some lay like crazy then basically stop after about 2 years. Others lay less but longer. Maybe they'd be happy to get rid of a noisy poor-laying hen who makes the neighbor unhappy.
Their responds depends a lot on how the owner feels:
1 about keeping the chicken project going,
2 about their individual hens, and
3 about you. And
(4 about eating chickens, but that's getting into dangerous territory for you to bring it up. Maybe best mention "rehoming" instead of "get rid" or "kill/eat". People have chickens for different reasons.)
One suggestion i have not heard is fencing the free ranging hens AWAY from your side during the day.
Also, hens and even roos' noise levels and schedules vary a lot. My landlady used to have a roo who crowed at 2am, 4am, 5:30, etc. The 2 and 4 am crowing were harder to deal with than the firehouse nextdoor. And THEY made a lot of noise banging metal (repairing firetrucks?) Even late at night with blazing spotlights.
If the neighbor cares a lot about eggs and not so much about the individual hens, she might switch to a quieter breed. It would take half a year for the new babies to start laying, though. If the neighbor wants to be cooperative in this way, you might need to continue being patient for a while. Craigslist is a good way to find a new home for the older hens, before the new ones are ready to go outside. I see likely downsides to this plan. 1. Chickens added, maybe none subtracted. 2. Maybe new chickens are just as noisy.
If your neighbor's hens are inside at 530 am, the owners can likely add insulation that would block a lot of sound until they are let out at a later hour. Even a layer of shingles on the outside would probably help.
And here's a terribly horrible suggestion that is very unneighborly. Attract predators. This option to be reserved for much later, and definitely after conversation plus some months time for neighbor to act on conversation.