Well there are a few ways to get your database.
It looks like you need to SETTLE the dispute or start over.
1.Use PHPmyadmin on old server and dump database.---YOU CANT DO
2. Use admin menu in threads to do backup and then ftp in to get them in backup directory provided you set this up.
YOU CANT DO EITHER
3. Telnet in and get database YOU CANT DO.
4. Have them send you a MYSQL backup BASICALLY WITH THE DISPUTE HIGHLY UNLIKELY.
5.Remotely link PHPMyAdmin and dump database. Possible but HIGHLY UNLIKELY AS MOST HOSTS DO NOT ALLOW REMOTE LINKS.
Basically you are being told by your HOST WE WIN you settle or start your site over.
