If you setup the Contact form 7 and put in a junk or noreply email address to send the message to on Contact form 7 it will still log the message being sent in the contact form DB
So in the shortcode template send message to junk@yourdomain.com and whenever a user submits it will try to send to that email and will still log the email as being sent. Personally I would actually create that email address so that your smtp doesn't get flooded with bounce back from smtp server.