No problem. I don't see any reason why you have to change the database host names - they are all just "aliases" anyway. As long as the database hostname is in the DH DNS, and you are connecting to it from within a DH domain, it ought to work as long as your forum/other software config points to the right database (regardless of what it is called).
That said, to answer your question:
I don't know for sure what you mean by "switch" - there is no need to export/create new/import the data from mysql.site1.com to mysql.site2.com. You can probably leave it as it is, or if you want (to keep the site name "the same" between the site and the mysql host) you can simply add an additional "host" of mysql.site2.com to the database in question in the Control Panel-> Manage MySQL section. That way, you could refer to the same database in either way.
Just make sure you have the forum/software config set to a proper hostname that is valid for the database in question.
Does that make any sense?