Email Sent successfully but Sent item in exchange is not being updated when email is sent using mail.dll.
Some SMTP servers automatically put sent messages in the sent folder, but this is not a SMTP protocol requirement and many servers don't do that.
I don't know of any way to check if particular SMTP server does that or not.
If your server doesn't, you'll need to perform manual email upload using IMAP: https://www.limilabs.com/blog/upload-email-to-sent-folder-after-sending