Islamically it is not a requirement for the MIL and DIL to live in the same house especially if they do not get along.
However the son has a duty of looking after his mother, but that doesnt neccessarily mean living in the same house. He could visit often and bring food for her etc.
As annoying as the lady is, it is a blessing to keep an elderly in your home. Where is your sister's MIL going to go? Her other kids wont take her.
Tell your sister to grow up a little and be accomadating. And you guys should have done your research into this family, before getting people married into it.
Just because a family is "religious" doesn't mean they're going to cooperate with you and be reasonable. In fact, our Pakistani "religious" families are often uncooperative and VERY unreasonable.