I’m still fighting this and tried something a bit different. Instead of creating a standard (code) app service, I created a container app service. I can pull the image directly from Docker Hub and store the files in an Azure File Share and I have none of the issues we’ve been encountering with standard Azure Web Apps.
I actually wrote up some docs on how to configure that: Azure - PowerShell Universal
I really like this method because then there isn’t a need to setup deployment of the application bits and you can just setup deployment of the configuration files to the file share.
That said, there is a caveat. Windows Containers do not support a Basic tier and their Production tier is super expensive.
Linux Container App Services do support Basic Dev \ Test tiers and are free or very cheap.
They also offer more reasonable Production tiers for Linux.
So if your scripts can run on Linux (or in the Integrated mode), then this may be an option. You could also run Windows Containers if you have deep pockets.
I’m still working on figuring out what’s happening with the storage on Standard App Services. They do not support path mappings to file shares like containers do.