Nop, there’s no way to add a database at localhost, becuase Dremahost has special servers set up to handle all of the databases.
You could go for a dedicated server, and then you’ll have databasese at localhost.
Your best bet will be to set up your coding differently. It should be easy enough to differentate the mysql connection from any coding you’d be changing. If you need help with that, post your code or explane what you’re trying to do and I’m sure someone here will be able to help out.