Can a username and password be hard-coded in a system DSN
A System DSN is stored in the registry and does not have the ability to store a password.  If you want to store the password, you'd have to use a File DSN and that would require changing the CRM to use the file DSN which you say is not possible.  Usually applications that use a System DSN, rely on SQL Server authentication of the users for their security.  When the application opens it should prompt the user to login, and this is the username and password that is used with the System DSN information to connect to the database server.  If the DSN uses Windows authentication and you are eliminating your Windows Domain, you are stuck.  You would have to change the CRM to make it work a different way.