I have just had to read your post through a couple of times to get it right in my head!
By saying that your BiL is not permitted to parent the child, I take it that he is not allowed to discipline her?
The child sounds as if she is desperately confused and wanting constant attention - or there is something going on that you are not aware of that is causing the wetting and other behaviour.
There is nothing old fashioned about expecting a child to behave in a certain manner in certain places. All children need boundaries and this child obviously has none.
As far as your MiL is concerned, I would guess that she wants Isobel to feel she is part of the family. It is hurtful that the pictures of your family have been taken down, but they are only pictures, doesn't mean she loves you all any less.
As far as the fact that this child seems incapable of sharing - many only children have to learn how to do this, it isn't wired in - you only have to watch groups of children at nurseries to learn this!
With the toys, I'd suggest that your children take their own toys to Grannies and you make sure that they are packed up and returned home with you each time. Bit of a nuisance I know and if you spot the child trying to take one of the items home with her, you need to say calmly " X and Y have been very kind sharing their toys with you, but it is now time for them to take them home . You have toys of your own to play with at home, these ones belong to X and Y, they are not yours!" If she starts to scream, have a tantrum, wet herself, take her clothes off or hurt your children, leave.
I don't understand the walking on eggshells but I can fully understand the different way that the MiL treats your children - my own mother did the same with how my son and my brothers twins were treated. I could only put it down to the fact that both of the twins had been seriously ill before their 1st birthday (one had heart surgery, the other had brain surgery) so she cherished them a little bit more.
Angela Rayner cleared by HMRC. What a coincidence!

Children sometimes get attached to a particular toy and it's understandable they will get upset if it's taken away. 