Well, I'd suggest trying to 'bump into them' accidentally some night... if possible.
If there is no possibility you can accidentally 'bump into' your friend and her HB friend in real life, then I would almost go 'shameless' on this one... Since - yeah - you don't even know this girl... have never met her, and probably won't ever meet her (particularly if Facebook game is your strategy - that's a rough technique and hard enough even when you know the girl)...
By 'shameless' I mean -- I don't know how well you really know your FB friend, but if you have the type of relationship with her where you can make a joke, send a text or talk half-casually with her, you could try something like "who the HELL is that girl in your FB pic?"
Tell her she's hot as hell and ask if she's single. That's enough that, if the girl actually is single, word will get back to her you're interested...
Change your profile pic to the best picture you have... and curate your feed a bit. Remove the drunken pics and awful photos (or hide them)...
I dunno - that's a bit of a hail mary.
Something like this worked for me once (before there WAS a Facebook, however - I'm old).
What do you have to lose? Dignity? Well yes there's dignity... but you're already running FB game, so that's gone anyway
