You say to DH, I know you feel sad and guilty that Mum is now on her own, but that's the way of it, she's going to have to learn to live on her own, and the sooner she starts the better. She needs to start building her own life without DH or her mother, and she needs the boot in the backside of living on her own to get out there, make friends, join clubs etc.
Your Mum has made life a lot harder, and very uncomfortable for me in the past, particularly when we were living under the same roof, and I'm not putting myself in the position where that could happen again. She openly favours the DD's, and is unkind towards DS, and you owe it to your son not to invite that kind of disparity and hostility into his home, the place where he should feel safe and loved.
So no, she's not staying here. She's not living here. We will of course, spend appropriate amounts of time with her, and occasions, but not to my detriment, and not to your children's detriment.
If you feel your Mum needs a sleepover, YOU can have a sleepover at your Mums, and if the DD's are okay with it, they can go with you. But it ain't going to be here, it ain't going to be me, and it ain't going to be DS doing the sleepovers.
Your mother made her bed. Now she can damned well lie in it. HER bed. Under HER roof.