Well, I have good news and bad news. I have successfully tracked down the exact issue.
If you are using Apache 2.4 with mod_ssl, and mod_ssl supports both NPN and ALPN the SSL negotiation will fail. None, or NPN only however, will succeed. This issue may reoccur in future with other sites that also run on HTTP 2.0
Thank you for your help, but as this issue appears to be in a support library, and not actually in Postman SMTP I'll mark this as resolved.