Hey there! I'm not finding much on my end on creating an outbound remote server for just one domain. We can do it for an entire machine, but not for just one account. This sounds like it would be a great feature request if you'd like to submit one using the link in my signature, and I'll make sure it gets reviewed.
Since this is just for one domain, could they not adjust the MySQL configuration files in their application to connect to the remote system? For example, if they were using WordPress, there is the DB_HOST value where they could specify the remote server. The remote system would just need to accept the connection and the user/pass would be created there. I'm just not seeing a good way to get cPanel tools (like the cPanel >> MySQL Databases interface) integrated with a remote database for just one account.