I will try to use the API directly to test if it works.
dotnet publish is not the command to publish a nuget package. It is use
to restore nuget dependencies, to build a project and to copy all required
files to a separate folder.
Next to your provided link there is a documentation how to push snupkg files. 
And as you can see here, it depends on SymbolPackagePublish resource found in the service index.
But this resource is missing in the service index delivered by our proget installation. Maybe this could be
a hint for you which can help solving my issue.
I will let you know if the direct usage of the API solves also my issue.
Many thanks for you support.
Best regards
Christian