First of all, I think the part about letting your daughter spend time with her grandmother isn't making her deceitful. Confused, certainly... and under the circumstances, who wouldn't be? Plus, she's getting carte blanche from Grandma that the boyfriend is okay (because Grandma's being deceitful toward you). Confusing, again. Grandma is also exploiting and sanctioning their spending time together without your supervision, and likely trying to win your impressionable daughter over. All she has to do is say a few right words to make your daughter feel like she's really in the right. It's sad.
Your daughter has been put between a rock and a hard place, because you are trying to make boundaries with your mentally ill mother through your her. Your daughter is not emotionally mature to do this dance with both of you. She wants to be the good granddaughter-- what granddaughter doesn't? Imagine the amount of control, resentment and manipulation which your mother levels at you...she might now be directing it toward a young child who has no idea how to gracefully and confidently tell her grandmother "you know, I'd prefer not to discuss Mom's business".
I've seen what comes of this in my own family of origin. The results aren't pretty.
I do think you must either supervise contact or deny your mom contact with your daughter. Your daughter is too young to understand all the elements in a relationship such as this. (This question, to me, is almost like one asked on a forum for children of narcissists or borderline personality disorder sufferers. ) I have seen the grandchildren of people in my own family deeply scarred because of people like this.You are going to have to step in. It's important to know what is being said, and your daughter needs you not to put her in the middle. You need to answer or rebuff your mother's questions yourself, and being present will give your daughter the support you likely wanted at her age. Have her back. Don't give her opportunities to break the rules with grandma, either. She needs you. Your mother's situation is sad, however, she's an adult. It's the child that needs protecting.