I'm sorry to hear about your Mom.
In this case, the dog obviously cannot be rehomed. People need to come before animals and if you Mom needs this poodle, that is understandable. I still can't say I would do the surgery because there is no guarantee that the neightbors couldn't hear the bark unless the procedure has changed since then. Have you tried a trainer?
So my first post was wrong. I might consider it in this instance. I spoke too soon and I am sorry.
I certainly would never want to lose Ellie.