Well L.,
I'm praying for you. You are stuck in a situation that is very difficult. You are sharing your child with someone who seems to be caught up in being the "cool" parent with his young girlfriend who is also "cool".
Remember, you are her mother, you have primary custody. He may be her father, but you are not in a popularity contest. Parenting is tough and you have to make tough decisions. You are very nice to let her keep this account and he needs to know that if facebook knew her age, they would delete the account. You are not being the "bad guy", by their standards she is too young to have an account and this is for a valid reason. They know that a child of 6 would have difficulty in managing their profile information as private and could easily be lured by a deceptive adult online.
With that said, her profile is private and that's great and safe, but wouldn't it have been nicer for him to ask you to set up an account that you could have her use or just have her use his account instead of making one for her without asking your consent or lying to an online website about her age?
I am a very tough parent and I have my reasons. My child was hurt by a predator many years ago and because of that experience I have had my eyes wide open to the dangers surrounding children and in this world. I pray no other mother or father would have to go through what I have been through, but statistically speaking it happens every day all day throughout the world.
L., you and every other mom that reads my post. NEVER feel that you are too paranoid, or that your kids will hate you forever because you didn't let them do what they want. Keep an open line of communication, love them, try not to judge them too much, BUT be tough in a tender and loving way for their own good if you must and don't feel guilty about it. There will be others that may come in their life that will want to be the "cool" person in their eyes. Just remember those tough decisions you have made are for their own good and are the reason why they are safe and not sold on the black market as a slave, or worse. Seriously, take it from me, there are people out there that are masters at gaining your trust or that of your child.
Don't feel guilty!
I'll pray for you!!!