>>
>>The connection strings are usually in the config file so just a question of using different configs for the two databases.
>>Find the EntityContainerName in Edmx properties and look for that in the configuration/connectionStrings section.
>>The actual connection string for the SQL server is contained in the connectionString property (provider connection string)
>
>Can I change config at runtime using the same .exe? Or would it entail 2 different builds? (pardon my ignorance).
>
>Alternatively, I think I know where to find the connections strings now, but I'm not sure how I can "inject" this at run time.

I usually create a custom configuration (although you might be able to use the standard 'Debug' and 'Release' configurations) and use web.config transforms. Little bit of work setting up but well worth it once in place since you only have to change 'common' settings in one place. See:
https://msdn.microsoft.com/en-us/library/dd465318(v=vs.100).aspx