mwheelan,
This can be done by modifying the smtp_banner directive in exim.conf. Do the following:
1. create a file, say /etc/smtp_domains and add the following lines:
primary_shared_ip: server_name
sec_ded_ip: domain_name
eg:-
66.67.68.69: servername.com
66.67.68.70: customerdomain.com
2. Edit the smtp_banner section in exim.conf and change it to the following:
smtp_banner = "${lookup {$interface_address} lsearch {/etc/smtp_domains} } ESMTP Exim ${version_number} \
\#${compile_number} ${tod_full} \n\
We do not authorize the use of this system to transport unsolicited, \n\
and/or bulk e-mail."
3. Restart exim and done.
Do a 'telnet customerdomain.com 25' now and see if you are getting the domain name instead of the server name in the SMTP banner.