How to explicitly name the database when using Entity Framework Migrations 4.3

You can avoid managing it in app.config by offering it as a parameter:

Update-Database -Verbose 
 -ConnectionString "CONNECTIONSTRING" 
 -ConnectionProviderName "System.Data.SqlClient"
 -StartupProjectName WEBSITE_PROJECT -ProjectName MIGRATION_PROJECT

Easy-piezy, if you love to type endlessly.