One possible strategy could be to give her a very specific, and modest, amount of "free storage" space, like a toybox or drawer or set of drawers, where she can store whatever she likes - of course, with the exception of anything dangerous, decaying, or stinky. In that space, she can keep her gum wrappers and sticky notes and whatever. But anything that overflows that space, you throw out.
That way, she has the power to keep _some_ stuff without having to justify it. She gets some practice with setting priorities, and dealing with limits, on stuff, and her stuff isn't holding the whole rest of the house hostage.
Now, she may be unable to set those priorities, so you may just have to fill the free storage space with what fits and throw out the rest. But you are, at least, giving her the opportunity to make choices.
And, come to think of it, the choice to tear off Barbie's face and keep that one small portion of a larger hunk of packaging _was_ a setting of priorities. I think that it's a good sign. She doesn't need to make the "right" decisions by any arbitrary standard, she just needs to make decisions that allow her to pare down the stuff to what can reasonably be stored.
I'm not suggesting that you put all of the storage space in her room under her own discretion - as a parent, you have a right to decide that even if _she_ says that she wants to keep that cubic foot of gum wrappers, her Barbie collection will have more play value. I'm proposing a specific, limited area, preferably one that can be closed so that there's no arguing about stuff piling up on top and falling over.
Is it possible that she's engaging in this hoarding behavior to please her father? Hoarders tend to model the idea that keeping everything, and having excessive sentimental attachments to things, makes one a good person, and that decisiveness in getting rid of things is a characteristic of a nasty, cold personality.
As the child of a hoarder, I know that I _still_ feel as if I'm being a combinatin of rebellious and holier-than-thou when I clean the house. I clean it anyway, but I can almost feel, across the miles, my mother's contempt when I do. (There's a reason why there _are_ so very many miles between us. :))
So I could imagine that your daughter may fear that if she confidently gets rid of her stuff, her father may feel that sort of contempt for her. In fact, she may feel a need to make a show of fighting and protesting when you get rid of stuff, so that he sees that she has proper warm, kindly, good-person hoarder characteristics. Yes, I'm being sarcastic about hoarding being a characteristic of good people, but it's possible that your husband and daughter _do_ see it that way.
If so, it would probably be a kindness to take a lot of the decisions out of her hands - if her father is going to express contempt for someone for not saving gum wrappers, it's better that he express it for you, not her. It's still _lousy_ that that idea is even in the air, but the more she can be shielded from that sort of dysfunctional judgement, the better. So I'd still advocate the free-storage space, but if she's afraid to decide what goes there, for fear of what her father might think or because she fears that setting storage priorities is a characteristic of a bad person, then I think that it would be best for you to just decide, for her, what to toss.