How to handle a clingy MIL

So my MIL expressed concerns when I was first pregnant about not having much of a connection with her grandchild because as I have my own mother, I would seek advise and help from her before my MIL. I understand her thought process, and I must admit since having my daughter I have spent more time with my mum because well...she's my mum 🤷🏽‍♀️ and I need help with breastfeeding and things and I feel more comfy with my mum around rather than my MIL, so my daughter sees my mum a lot more. Anyway, to help soften my MIL I update her on everything, I send her pictures nearly every day and I ask for advise here and there, even if I know what to do just so she feels more included. We visit her twice a month too.

Anyway we were recently out at a family Christmas party and from the second we arrived she just wanted to hold our daughter. She didn't ask first she just took her out of our arms when we weren't looking. Our baby is super fussy in the evenings and only wants mum or dad, mainly mummy but we are trying to encourage her to enjoy her daddy in the evenings, but my MIL wouldn't listen and she kept crying. My partner kept taking her back everytime this happened and I could see my MIL becoming more annoyed, eventually she stormed off for ten minutes and my BIL had to chase after her. He told me later that night she was frustrated my partner wasn't giving her time to soothe our daughter, using the typical "I've had 4 children I know what I'm doing" line. BIL told her all babies were different and that we know our daughter and how to comfort her more than anyone else.

She has a real separation issue too, when we have to leave she sometimes won't give our baby back, jokingly trying to pull her back, which infuriates my hubby because he then has to raise his voice and demand his own child back. The two of them clash A LOT over this and I have no idea how to settle it. They both suck at communication too, but I can tell she wants more nanna time, and he wants her to ask permission before snatching our daughter up and to not act like she's parenting her.

I don't feel comfortable leaving my daughter alone with her and I EBF so I don't want to leave her with her for a few hours to babysit because I know my baby won't be comfortable around her. My daughter is very picky with who she will let look after her, rightfully so because I wouldn't feel comfortable being alone with someone I hardly know too. She's only 12 weeks.