How to use a dynamic connection string using a Model First approach but still use the data model in the EDMX?

I have created an EDMX using EF 5 with the Model First approach, i.e. I started with a blank designer and modeled my entities. Now I want to be able to use this model defined in the EDMX but supply runtime SQL Server connection strings without modyfing the config file.

I know how to pass a connection string to the DbContext but the issue is locating the metadata for the mappings within the assembly.

For example, my EDMX has this connection string in the app.config

<add name="MesSystemEntities" connectionString="metadata=res://*/Data.DataContext.EntityFramework.MesSystem.csdl|res://*/Data.DataContext.EntityFramework.MesSystem.ssdl|res://*/Data.DataContext.EntityFramework.MesSystem.msl;provider=System.Data.SqlClient;provider connection string=&quot;data source=MyMachine;initial catalog=MesSystem;integrated security=True;MultipleActiveResultSets=True;App=EntityFramework&quot;" providerName="System.Data.EntityClient" />

The part that I am missing is the "metadata=res://*/Data.DataContext.EntityFramework.MesSystem.csdl|res://*/Data.DataContext.EntityFramework.MesSystem.ssdl|res://*/Data.DataContext.EntityFramework.MesSystem.msl;"

I want to be able to create a DbContext programmatically passing in a SQL Server connection string but "add on" the metadata portion.

This is what I would like to be generated by the T4 file...

public partial class MesSystemEntities : DbContext
{
    public MesSystemEntities()
        : base("name=MesSystemEntities")
    {
    }

    public MesSystemEntities(string sqlServerConnectionString)
        : base(GetEfConnectionString(sqlServerConnectionString))
    {    
    }

    private string GetEfConnectionString(string sqlServerConnectionString)
    {
       // values added by T4 generation
       string format = "metadata=res://*/Data.DataContext.EntityFramework.MesSystem.csdl|res://*/Data.DataContext.EntityFramework.MesS