So it seems that once you run/debug your site, the data provider attempts to look for the connection string in the Web.config of the main project. So I renamed the app.config in my F# project to Web.config, and made sure to have the same connection string listed in the Web.config file of the main project as well. That way I appear to have access to the schema while coding, and when the project runs it picks up the connection string from the main project. This is a working solution for me, but if anyone can shed any more light on this, please do so. The new syntax is:
SqlDataConnection<ConnectionStringName="dbname", ConfigFile="Web.config">