Configuring AppSettings with ASP.Net Core on Azure Web App for Containers: Whither Colons?
After more experimentation than I'd like to admit I think I have the answer.
Where you use : on an App Service, use a __ (double underscore) on an App Service with containers.
So Parent__ChildOne instead of Parent:ChildOne.
To read a little more (not much more), I wrote this up as a blog post here: https://blog.johnnyreilly.com/2018/07/28/azure-app-service-web-app-containers-asp-net-nested-configuration/