FedEx :)
Sorry, very old joke among system admins. Some anonymous client back in the early days of the net went to an anonymous consultant with this request:
"I have 650 megs (See? Very old joke. I can put that on my iPod Touch without issue.) of data that I need to back up from my office in City A to my office in City B each and every day. What data transfer method and/or data connection would you recommend?"
Answer: FedEx.
But anyway...
When I was on CPanel, we were using a "bug" to help us out here. If you know the name of another client's database along with their MySQL username and password, you can access that database from another account on the box. Our method was to place the MySQL database(s) into a separate user account, use the information from that second account to allow the main account to access it, and then just do a complete account backup on that second account to get the databases. Since the second account only had the databases, it was easy to do.
It's a bit of a concept to understand but it worked well for us.
Some other backends allow this while other hosting backends are *cough* more secure and don't allow this.
I see it as a bug myself but since it;s present, it would be foolish not to use it if is helps.