A good rule of thumb here is that if they are too toxic/difficult for you to deal with, then they are certainly too toxic and difficult for your both vulnerable and defenceless child. Your child does not need or warrant such a poisonous and negative person for a role model in their life, people need positive role models.
Your reason for maintaining any contact at all with such a person is shaky to say the least (societal convention) and is really no good reason at all to maintain any contact with. That only is viable if the grandparents are actually emotionally healthy which is clearly not so in this case. And no, she is not very very good with your DD either: how could she be if she is awful to both her mum and dad as people?.
Some grandparents really should not be allowed any access to their grandchildren. percentage of the general population is dysfunctional and/or abusive. That percentage, like everyone else, has children. Then those children grow and have children of their own. The not-so-loving grandparents expect to have a relationship with their grandchildren. The only problem is, they’re not good grandparents.
Many adult children of toxic parents feel torn between their parents’ (and society’s) expectation that grandparents will have access to their grandkids, and their own unfortunate firsthand knowledge that their parents are emotionally/physically/sexually abusive, or just plain too difficult to have any kind of healthy relationship with.
The children’s parents may allow the grandparents to begin a relationship with their children, hoping that things will be different this time, that their parents have really changed, and that their children will be emotionally and physically safer than they themselves were.
Unfortunately, this is rarely the case, because most abusive people have mental disorders of one kind or another, and many of these disorders are lifelong and not highly treatable. (Others are lifelong and treatable; however, many people never seek the necessary help.)
The well-intentioned parent ends up feeling mortified for having done more harm than good by hoping things would somehow be different — instead of having a child who simply never knew their grandparents and who was never mistreated, they have an abused child who is now also being torn apart by the grief involved in having to sever a lifelong relationship with the unhealthy people they are very attached to.
Do not let yourself be that well intentioned parent.