No, I don't think RC stores that in the DB, but queries the IMAP server for the number on each page refresh.
And even if RC did store it, the number would be stale if the user was not logged into RC, as there is no way for RC to fresh it as new mail arrives without a login.
To get an accurate, current count for your other site, you'll probably need to figure a way to query the IMAP server itself as well. You might be able to leverage the RC framework and code to do so, but I doubt that the job will be trivial.