I have been on both sides of this OP.
My DS has had a very disfunctional start in life and sadly has been both bullied and been a bully.
There is one lad in particular that made DS's life hell, and I know a lot about his own troubled back ground.
My loyalty was always predominantly with DS, but I have also tried very hard to be a responsible adult and be clear with this lad that he was welcome in our home as long as his behaviour was appropriate to DS, as at the core of the relationship, they are friends.
Of course any parent will feel protective of their own child, but I also feel that we can share a role in helping children with difficulties more than sometimes we perhaps do.
But on the other side, I have been very realistic with DS when it is him that was in the wrong, and spent hours talking to him about how people feel just the same as him, as he seemed to struggle for a very long time with empathy.
A child with Emotional issues is in some way one with SN, they both have to learn how to 'function' in groups, but also perhaps be given the space to learn social mannerisms that a lot of children just take for granted. Because this Little girls issues will certainly not improve if she is ostracised.
Perhaps without going into detail you could express to these parents that you are aware that DD is struggling to fit in, but that their patience would be appreciated.
My DS and this boy are now very close friends, DS always had the say about how this lad fitted into his life but I am glad I stuck to my guns, because the lad's life is still very troubled, and he knows he is welcome at ours any time, as long as he respects how we 'behave' at our.