Coping with a split household and mental manipulation

I have a 5yo (almost 6) daughter in which her mother and I have split custody almost entirely right down the middle (without over complicating, we essentially have a rotating weekly schedule). My daughter has confided in both my fiance and myself that her mom gets really REALLY mad if she even mentions what she does at our house but will also just go buy her whatever it is she's talking about at my house. We bought a trampoline, my daughter told her mom, her mom GROUNDED HER for telling her and then bought her a trampoline the same week. Also, "My mommy said I have to let you know how much I love her" and "My mommy said I have to love her more than you". The last two quotes were this morning's conversation. This is just the tip of the iceberg but characterizes the situation well.How do my fiance and I help my daughter learn that this kind of behavior isn't okay? How do we, as parents, cope with this? I keep telling myself and my fiance that empowering her with knowledge and compassion is the best course of action here and that she will see through the facade eventually but I'm worried it's not enough. via /r/Parenting https://ift.tt/3cS2z18