I think that in life, the person that you should ultimately please is yourself - because if you try to please everyone else, you get so frustrated and tired of everything because you are not happy. And for a parent, that's what's most important because you have to be happy and well-rested to look after your family!
Having said that, I think you should not move to your MIL's place just to please her - BUT, you can make other arrangements like going over for a weekend or smth! I think the main thing is that your MIL feels like she's missing out on her grandchild's life, so she wants more time with her - so perhaps going back as a family once a week, or planning outings that bring your PIL together would help :)
Of course, you need to explain clearly and calmly why you chose not to - and I think that having to take care of the house and her grandchild would be too much for her to handle.
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