I am building out a web solution for a client with an option to send emails of the sites content to others.  The site is hosted on our system, as well as an SMTP server just for this feature.  However, in order for our SMTP server to be active, there is an MX record pointing to us.  Problem is, there is an organizational server in the clients network that handles their emails.  Our web solution is trying to send out from the SMTP on our end and it's finding itself first when the client trys to send to internal employees, instead of finding the organizational server.  The client cannot give connections credentials for their server for security reasons.  I have tried a forward on our side, but it hasn't worked.
Email sending from server outside of network
email;exim