I just went through this with IdeaBlade support. Tried to do exact same thing. And their code sample shows how you get connection string using regular EF4 context :)
This is response I got:
I'd suggest that
instead of trying this approach, you actually created the EM after entering the
login information.
i.e.
1) Login UI: user enters accountID, username, and password. Click login.
2) create EM: var manager = new IDATTApplicationEntities(true, accountID);
3) login: manager.LoginAsync(credentials);
Basically, after you figure out that you need to pass blank EM into constructor you will get into limitation where your DataSourceKeyResolver won't get keyExtension