The MX record should be pointing to your own Mail server. If you create a new MX record pointing to an external IP I fear that it will mess up your internal email as well. Just a thought but ain’t sure and perhaps you can have a try. If it works then great otherwise just take it off again and things should be back to normal (make sure DNS has been replicated)
Another way to resolve this is to create a POP3 account in Outlook and use that account ONLY to send email to this particular company. Make sure you enter your SMTP server name (the one given to you by ISP) and NOT your mail server name in SMTP Server. Also need to open Port25 on the fireware (router) to allow email being sent from clients.
However I do not favour this solution. I’d rather rename my domain name.
Depending on how you have DNS setup, you can have SPLIT DNS such that there is an internal MX record pointing to an internal host/IP. The external MX record can point to a public IP for a firewalled internal host.
After further reading…. the only role DNS plays in this situation is routing the outbound SMTP traffic. Inbound sounds like a pull event. For outbound SMTP relay, you should be okay with the same internal and external domain names as long as the host part of the name is unique to each side of the network edge. For example, the ISP’s SMTP relay may be called smtp-ob.abc.com and the internal hosts would have names like mailhost1.abc.com and mailhost2.abc.com. There should be no fear of DNS name collisions if you set things up along these lines.