My mom stayed with me when I had my first. It was a blessing!!!!!!!!! Too bad she had to go home (25 miles away). That's when I began having too stay up all the time. Lack of sleep. She did come out every now and then so I could get some sleep.
Make the offer. You want to be there for your daughter and grandchild and yes SIL too. Even if it's so she can rest some. If SIL says no explain "A happy mom is the best wife and if she's tired it's like she's having PMS." LOL
It's also bonding time with grandma!!
GO FOR IT! ALL they can say is no thanks.