The best way to do this is via the customer's PHP container. You can either "su" to their website user from root or add your SSH key to their website under "developer tools" and connect to the server with SSH.
Once in their environment you can use the MySQL CLI:
mysql -u their_username -p their_database_name < your_backup.sql
You don't need to specify a host with -h unless the site has a remote MySQL.
Enhance doesn't install the MySQL CLI onto the host o/s. If you want to be able to import backups as root then you can install it via apt however I don't recommend this.