If you have multiple domains on your server, it is fairly easy to move one of them to your second IP address. You would need to create a new DNS zone for the domain using the secondary IP address, then create a virtualhost entry for the domain in httpd.conf using the pre_virtualhost_global.conf include file. This would create a dedicated IP for the domain and would allow you to access the "homepage" via the domain name or via the IP address. I've done this on my server and it works great, though to be honest I did it for other reasons...the dedicated IP address was a bonus.
NOTE: I don't think the above will work with the primary domain associated with your account. I suspect you would need to change the primary domain to one of the other domains on your server, then make adjustments in WHM/cPanel to relect the change in primary domain, before you could perform the above procedure.
If you only have one domain on your server, I'm sure it's possible to edit its DNS zone to the alternate IP address, and create a vhost.local file that edits the main virtualhost entry to use the alternate IP address. If you're fluent in PERL this would be a fairly easy task; otherwise, it's probably not worth the effort.