A troll score could probably be constructed by Facebook and Twitter, but potentially they are not very likely to want to do it. Any suggestions on how to get around this problem would be appreciated.

I don't see why this wouldn't work fine as an optional feature. You could even have minimum (or maximum) positivity ratings by group, say--someone might want to see everything their core group of friends and family posts, only positive things that acquaintances or strangers post, and only negative things posted by their frenemies.