This is not unreasonable at all. As a matter of fact, it is actually agood idea, especially with any family member who is not so good with boundaries. New baby, sleepless nights, hormones and potentially over-bearing family staying in the house with you are not a good mix at all.
When I had my DD, we told everyone that no one, absolutely no one, would be coming to stay with us for three weeks after the birth (both my family and the ILs were in different countries to where we were, and would have had to stay with us when visiting). I think everyone was more or less fine with that (I think, MIL protested, but was quickly shut down by FIL and DH), and it worked out very well. After three weeks I was getting the handle on the breast-feeding (it took both DD an me a bit of time to learn), and I wasn't quite such a banshee-howling bundle of post-partum hormones anymore.
As for the OP, the MIL is the one protesting, it should definitely be DH handling this and enforce the rule. If OP is alright with MIL staying at a hotel and visiting during the day, that might be an option if MIL agrees. But as the OP has stated that MIL has problems with boundaries in general, she might not...