I’m not sure what I would do in your position, but I think i’d lean towards doing what needs to be done to delete the account, even at the expense of providing additional PII. I wouldn’t do that for any old account, but for something like Facebook, where its pretty likely that they (1) know who you are already (2) already have your face and name, (3) probably have much more about you based on your posts/activity, and your friends posts/activity and address book.
In that context, I think the cost/benefit skews more towards getting the account taken down. Especially if you live in a jurisdiction (EU, California, various other US states) where you have a legal right to have your info deleted).
edit: I would probably submit the ID with additional details (address, height/weight) covered up, unless they specifically said not to (I’ve done this with ID verifications before, and not been challenged)