Creating a Linked Server that Points to Itself
Turns out that I was able to get it working with different parameters.

EXEC master.dbo.sp_addlinkedserver
    @server = N'LinkedServerName', 
    @srvproduct=N'', 
    @provider=N'SQLNCLI', 
    @provstr=N'DRIVER={SQL Server};Server=(local)\InstanceName; Initial Catalog=DBNAME;uid=user;pwd=password;'