Mike, define them a new company policy of yours, to keep back-up of Moodle (together with DB, moodledata and moodle) folder on site. (This is one possibility)
The one you're talking about, migrating DB, I would think about using it as last resort, as in most cases, database resides locally on hosting servers, and possibility of giving remote access to transfer DB over will may become hurdle, unless you are setting up server in house, then it can be a possibility.
Even if they haven't provided you with CPanel that you can look into your account files, which may look like the reason, as CPanel cost licensing fee, the least they could have done is given FTP access to the root of your account, additionally given you PHPMyAdmin to access your database, both of which are free to implement, and I dont have any doubt if given a valid reason, they can' provide you with both.
And reason can be simple, you dont even have to define a new company policy, just tell them that you have to manually upload some data into moodle and moodledata folder and for database, you need access to look into tables. This way you can backup both without issue and giving them a clue that you have plans to migrate., and they shouldn't have any issue with that as well, for both FTP and PhpMyAdmin isolates your account with others residing on the server.
And if they are so sensitive in giving access to customers, then as Emma said, backing up courses and restoring them into new moodle - this will give your moodle a fresh start as well.
