I have a site running n Azure, which uses EF Database First model.
If I embed the connection string in the web.config and deploy all is rosy.
The problem is not the connection string, but the inability to specify the providerName
from the portal.
Try publishing with a local or dummy connection string in place, and overwriting it as you have in the Azure Application Settings. See this answer: https://stackoverflow.com/a/18964294/94853