So, your MIL is hurt, so she recognises there is a problem - she just needs to recognise that the problem is with her, and you to a certain extent.
My DD is 4 and is frightened of my ILs, we dont see them often but when we do go there - MIL insists on getting in DDs face, i feel like saying, if you just leave her alone and stop treating her like she is the main attraction in a freak show then she might warm to you a bit, but how can you say that to someone who is just being kind and wanting to cuddle their GD?
But you could just say "why don't we try......."
Tell you what you could try, which goes against the grain a bit, but how about leaving DD with her, just for an hour? I know that my DD can be a bit funny with my mum sometimes, my DD is very clingy (not suggesting yours is) but take me out of the equation and she settles down and they have fun together.
If your MIL were to come and chat to you, ignore LO and just wait til she goes to her or sits with a toy in her lap or something, and not force herself on DD.
Once a week and she wants more? that is UR, she needs to let you get on with your lives