Forward mail from Exchange to external host in addition to local delivery

I've just created another send connector (Internet) and chosen Smart Host to send (SMTP server). After clicking next there was basic authentication settings which I've filled in and it seems to work. Mails are routed thru this SMTP server.

Something similar came up in this question, and I wanted to mention something I left out of my comments there:

This is a fundamentally kludgy thing to do, and no matter what you come up to make it sort-of work, it's gonna be a kludge.

You're taking two mail systems, setting them up as if they're the mail system for yourdomain.com, and then trying to get them to talk to each other using mechanisms that are intended for moving messages around between different email domains. Of course there are problems! (Don't take this personally, BTW, I've had to do it in the past, albeit on a more limited basis.)

One kludgy idea: I don't use Google Apps, can you set it up to receive email for another domain (yourdomain.net or temp.yourdomain.com) and deliver those messages to the same mailboxes? i.e. ugly@yourdomain.com and ugly@temp.yourdomian.com both go to the same user's mailbox? Then Exchange can copy any sent messages to that new domain. The ugly part with this setup or many variations of it is that people will get a copy of any "Sent" messages in their Inboxes.