Honestly, DSD would be better off without that constantly cycling disappointment.
Long term, it would be best if you don't do all the legwork for this woman, let her make the effort.
She won't. But that's HER choice. It doesn't mean a thing about DSD.
She still calls this woman 'mummy'? At her age?
My son hasn't seen his dad for the last 3 years (abroad) skyped a couple of times, called (well I called him for DS) a few more.
My ex was abusive to me, and therefore terrifying to DS. DS missed him for a while, he does love him, but they have no real relationship. He still calls him Daddy.
He's called me Mum since he was 4. Only Mummy when he's telling me he loves me. :)
He's SO much happier as a person without the shouty man in our lives, his confidence is light years ahead of where it was because he doesn't have some sub-human chipping away at him.
Your DSD may well end up being a future stately homer* if she carries on having hopes raised and dashed and no consequences for this woman's woeful psychological neglect of her child.
*The Stately Homes thread is a thread on here that supports those of us that have grown up with dysfunctional parenting.
You and your husband have to counteract all the harm this woman does, so give her as little space as possible.
Don't offer the photo, if DSD wants to give it, that's ok, but don't rush around or make any effort at all to make this woman look like a normal parent. Begin to allow the truth to show through, and manage the required distancing.
Your H needs to have a proper age appropriate chat with his little girl and tell her that this is not what being a mother is, and that it's ok for her to love her mother, but not to expect anything of her.
He needs to have the people let you down conversation with her, and reinforce how much he and you care for and are there for her.