Well, all calibre does is send a standard email containing the book, using your email address.
I assume the email address you entered in calibre is added to the list of emails that is authorized to send stuff to your Kindle account, since you say you can send emails directly via
that same email address 
and it works...
What you are describing would seem to be impossible.
Can I assume the calibre job log shows success, and that the email appears in the webmail interface as sent?