My first untested guess how it could work:
- Whitelist this URL (http://www.facebook.com/feder.ejgv)
- Bypass any request around the "facebook block rule" if the HTTP referer header is this URL
Then your end users may be abled to see this site. Though I could imagine that stuff like posting comments, liking, etc will not work.
What you could do then: Do a tcpdump on you work station while you are visiting this facebook profile, voting, etc. Maybe you find some information which is unique to all requests on this profile (maybe part of all URLs is the same?). Then you could create specalized rules to whitelist those requests.
It may not be possible due to the fact that you want your users to sign in. I noticed a few images on your site point back to http://www.facebook.com/photo.php* and the app for your contest to "View and vote for entries" points to http://apps.facebook.com/easypromos-premium/entries/6322. Only being allowed access to certain parts of Facebook getting the full experience of your photo contest may not be possible.
My best suggestion is to just play around with it, take fschulte's advice and then log in yourself to see what's possible and what isn't. You can modify the rule based on your needs.