Just wanted to follow up on this in case it helps anyone else...
Please forgive me, I write software, and am an "email administrator" (if you can call me that) by necessity.
So after some research, it appears that SMTP has no mechanism to put mail in a "Sent Items" folder. That is the job of IMAP. So our email clients that we use to create an email and send it, actually send the email via SMTP, then use IMAP to store a copy of the email in the "Sent Items" folder. I had no idea that my messages had to be sent to the server twice, using two different protocols, every time I sent an email. Now I know.
Really, you probably wouldn't want an option like I am asking about. If you mixed and matched how mail was sent (sometimes from an email client, sometimes from automated code), then when you sent from your email client you would end up with TWO copies of that message in your "Sent Items" folder. Once because the email client added it, and a second because this proposed setting would add it.
My solution is going to be, in addition to sending the email via SMTP, I'm also going to save a copy in the "Sent Items" folder using IMAP (using code). This should work fine and get me the results I desire.