You can only run one MTA on each IP address. This is because MX records, unlike SRV records, do not allow a port to be specified. Mails from other sites always come on the standard ports (25, 465 which is deprecated, and 587 for submission). Those ports must absolutely not go to different MTAs, or you will experience odd delivery problems, though really you only need to serve on 25.
None of this has anything to do with dovecot, which is an IMAP and POP3 server. It's all about the SMTP server in this case, or MTA, postfix. You can run IMAP and POP3 on nonstandard ports as long as you are prepared to communicate them to your users.
You should set up an MTA on a new server on the IP address, and have the two existing MTAs use RFC1918 addresses. Have the new MTA act as a relay for those two domains, and give it internal DNS so that it sees the RFC1918 addresses in the MX records. Ensure it can only relay to those two domains, and your problems will be solved. You don't need it to relay outgoing mail, and internal SMTP clients can send mail through the existing mailservers.
Set your MX records for both your domains to the domain name pointing to your single IP, and set your SPF records on those domains to allow the FQDN given in that IP's public DNS to originate mail for them. Ensure your NAT is configured such that SNAT of your SMTP servers uses that IP.
That's it! It should just work after that, without any IT governance issues at all.