I do not know much about cPanel, but I know how to transfer a database very fast to another server - if you have access to ssh.
use mysqldump with the proper parameters and chain it with ssh. so the database is imported while the sorce database still exports. no temporary files are used (except mysql internally ;))
mysqldump --user=user1 --all-databases | ssh targethost 'mysql --user=user2'
if you authenticate to the source server with your private key and ssh-agent, you could use the -A option of ssh to connect. So you do not need to care about authorization on target side. But keep in mind:
Agent forwarding should be enabled with caution. Users with the
ability to bypass file permissions on the remote host (for the
agent's Unix-domain socket) can access the local agent through
the forwarded connection. An attacker cannot obtain key material
from the agent, however they can perform operations on the keys
that enable them to authenticate using the identities loaded into
(source: man 1 ssh)
hope this helps a bit