Do I need an MX record to send email via SMTP if that server will not be receiving email?

No you don't need an MX record. An MX record designates where email for a domain goes, not where it comes from. Anyone using the MX record as a test to determine if the sending server is valid is doing it wrong.

What you should do is make sure you have an SPF record for your domain that designates this server as a valid sender for your domain.

No. You won't need an MX record to SEND... However, you may want to make sure that you have a reverse PTR record for that host if you're sending to the world. If you're only interested in sending to your Exchange server, you can either set the Exchange server up as a smarthost (relay) on the Postfix system or add a receive connector in the Exchange server with the Postfix server's details.