Earlier quoted context omitted.
Common plight these days. I've considered making throwaway facebook profiles for these sorts of things before, but there's got to be a better solution.
Why wouldn't you make a throwaway Facebook profile? My Facebook account was initially created as my legitimate account, but now I treat it as essentially a throwaway. It makes logging in to most random little sites extremely handy, and I see no measurable downside.
If you want to remain any kind of anonymous, it means logging into each facebook login-enabled site with a new, fresh fb account. Using one trash account might stop them from linking things to your 'real identity' (such as fb perceives it), but it still lets them link all of your activities together and build a profile. Persona linking from there isn't hard, based on usernames or email addresses used (see pipl and similar for examples of casual, publicly available version of these tools). But even if there's no persona/pseudonym linking algo going on, you're still being 'tracked'. It's not fully anonymous.