If FB couldn't figure out who you are, they locked your account and required a photo of your driver's license or other state/national ID to verify identity.
It seems unlikely that this process will involve a human to intervene. Bots could do this easily, and could exploit things your friends share without you knowing. "I got super wasted with x last night" being posted or...
Or worse. They get access to those Facebook Messages?
I could see this starting as an opt-in for a little bump (of rate, score, etc.) in your favor, like how the car insurance companies give you a discount if you let them log your driving and it shows safe behavior (e.g. https://www.progressive.com/auto/snapshot/)
You can create user groups or G+ style circles on FB. For instance, I've got posts that only my friends see, most of them are also visible to my parents and very few are visible to acquaintances.
Add the representative of the bank to one of the limited visibility groups.
Because _everyone_ has a facebook account? Bank/work somewhere else if the prerequisite is to have a facebook account that is accessible to people who are not your friends.
Alternatively, if you are genuinely worried about being able to do this, I would advise setting up a public profile separate from a personal one (cpt obvious).
I remember when it was a faux pas to put your real name on the internet - long live the days of having an alias that you can make a whole bunch of mistakes to a private group of friends without being publicly remembered for it.
A rare outbreak of common sense in US labour law, although I suspect Facebook also lobbied for it - it's against their TOS to give your credentials to the interviewer.
Could they do that? You'd probably have to read (and keep up with) the T&Cs.