I believe there is an issue with the DSN SRM is using to connect to the embedded database. The problem is that when I do a modify on SRM install, the DSN is grayed out and I can't change it.
Does anyone know how to change this seemingly "hardcoded" DSN value?
This only appears if you use the embedded database. You cannot change this value. See Modify a Site Recovery Manager Server Installation for more information
There is a command-line option to change the Database credentials that are part of the SRM installer. Post-installation, this can be run manually. I assume you just want to change the username with which you connect to the DB, so the syntax would be as follows:
C:\Program Files\VMware\VMware vCenter Site Recovery Manager\bin>installcreds.exe -key db:<DSN_NAME> -u <New_Username>
Obviously, the user/login will need to have been created on the DB already and have access.