We will have family visiting from overseas and this is the first grandchild in the family so a big deal. My mom is arriving 10 days after my due date and staying with us for a month. I just told her that we'd like the first week or two to bond and settle then she can come. We have a very open, honest relationship so I feel comfortable that I can let her know if I need some space or if she's stepping on my toes.
My MIL is a different story, she's extremely sensitive and emotional and I just could not deal with her staying with us in the first 6-8 weeks. She can't afford her own accommodation so we just suggested she come when baby is 4-5 months because it's a much more exciting time to bond and my SIL would be able to come with her so will be much better.
If you don't feel comfortable having your MIL there for the birth or directly after, that's your call to make. You decide and she needs to respect your decision. I disagree with the opinion that it's rude to ask her to stay elsewhere, this is not about your MIL, it's about you, DP and baby. It's a huge adjustment for anyone and you need to do what makes you comfortable. My dad is visiting with my step mom and sister when baby is 6 weeks and I have asked them arrange accommodation as I would have had my mom staying for a month before that and feel it would be too hectic, they can afford it so they were more than happy to do that.
You come first here, everyone else needs to fit in with you