You have a serious problem. You did three months of work without first establishing some key information about your customer's deployment environment, and without doing any sort of early proof-of-concept test. If your customer is sufficiently non-tech-savvy to have not seen the issue as soon as you argued for MySQL, it may mean they're relying on you to have verified that it's a viable option in their hosted environment.

The part of this that sounds off is that you were able to create a database through the hosting outfit's control panel. If it's the case that they support MySQL, but don't provide the pieces to access it, something somewhere is off. How do they expect people to access MySQL? PHP?

Assuming the company you're contracting for has some leverage with the hosting outfit, your best bet may be to apply pressure through your customer.