Mail routing relies on MX records in most instances. That's why Postfix is ignoring entries in /etc/hosts.
Do you have an internal DNS server for your domain on either of these machines? If so, adjust the MX and A records to use the machines' internal addresses. If not, you might consider setting one up.
I don't use Postfix so I can't help with that. In sendmail, you can override MX records with an entry in /etc/mail/mailertable. For instance, a mailertable entry like
would direct mail for firstname.lastname@example.org to 10.1.10.101.