Christopher,
Setting up an email alias forwarding to a CGI script should definitely be possible on Exim. Are you using cPanel? If so, you should be able to easily set up the aliases right in cPanel as noted in the
help.
If you're not on cPanel and this is
your dedicated server you might try adding an entry in /etc/aliases such as follows:
bounce@example.com: "/path/to/bounce.cgi"
then run: newaliases
I think /etc/aliases and newaliases is sendmail-specific, but I've seen it work on Exim, too.
If this is not a dedicated server your host should know the exact method to achieve email forwarding to CGI script.
There will be a way - it's just a matter of figuring out how!
Regards