Don't be chicken!
If your SMF/Coppermine combination is same as mine then they share the same database (for a common login) so in effect, it's even easier to move than two separate databases ...
Transferring your complete accounts is easy for someone with root access at this end — you didn't say whether you're on shared hosting solution,
VPS Hosting or
dedicated server? If the first, you can't complete your mission easily so open a ticket to Tech Support providing them with logins for all the remote accounts. If you're having a
VPS Hosting or
dedicated server it's easy to FTP to the remote server to retrieve a Home Directory backup of each account and install on your new server, complete and intact (including emails, stats, everything).