Previous versions of TB (v1 and v2) needed a password during the registration process. You were supposed to enter it during the online ordering process and were cautioned to remember it, because it wasn't stored by cifnet nor by ritlabs.
So your problem most likely concerns v1 or v2 and v2 has been obsolete for about two years, so I guess they couldn't help you anymore, though it would have been nice if they told you so.
You're talking about a new computer computer. Have you still got your old system with the software installed? If so then I can guide you through the mechanics of extracting your proper registration to your new syste, otherwise you're out of luck.
Of course you could decide to upgrade to v3, there used to be an upgrade discount, but I can't find that just now.

On your old system, you've got to open regedit (the registry editor) and got to:
HKEY_CURRENT_USER\Software\RIT\The Bat!
This is the part of the regigitry where TB stores all settings that aren't kept in the mail directory.
Currently I've got no system with v2 installed, so I can't give you an exact name for the key you've got to be looking for, but it's called something like
RegistrationID, there might be two of them.
That key should be copied to your new system and you're done.

When your message base is in the same location on both computers and the accounts have the same name, you can also copy the whole HKEY_CURRENT_USER\Software\RIT\The Bat! tree to your new system, that can be done easier, as you can export that to a file (File -> Export -> ....  In Regedit's menu) And that file can be imported in your new system.

You can automate the export part of this instruction by executing this:
c:\windows\regedit.exe /e c:\TBsetup.reg HKEY_CURRENT_USER\Software\RIT

Another way to restore all settings, but without editing the registry yourself, is to use TB's backup and restore function. That works reasonably well. The only problem is that it'll restore the old messagebase too and that it will do that to its original location. So when your new computer has its mail directory on a different place than your old system this will cause problems.
I'm not sure whether restoring an old message base via a backup will cause your new messages to be deleted or that the backupped messages will simply be added to your existing messages.