This is your daughter's fight, not yours. If anything needs to be done, your daughter needs to take responsibility.
Unless the order says otherwise, kids not yet in school follow the same schedule as if they were in school. (In fact, the order likely says that.)
The standard possession order says: "Either parent may designate any competent adult to pick up and return the child, as applicable. A parent or a designated competent adult shall be present when the child is picked up or returned."
Your daughter has picked you and the dad has picked his mom. There is nothing you can do about it. Be cordial. You do not have to let her in your home (but that will only serve to increase the tension.) Simply have the child ready and minimize the interaction.
Your daughter can file seek a modification to eliminate the mother as a "competent adult," but her chances of prevailing in that are not very good.
The mother should file a petition to modify and can request that the place of exchange be changed to a neutral location. She can alternatively seek to have the paternal grandmother disqualified due to her actions, which you have the right to record if you are present.
This is not legal advice and is provided for informational purposes only. NO attorney-client relationship or privilege is created, and none of this is confidential.