I agree that your mil's behaviour is a bit odd. On the one hand, when she visits you she could be taking your dd out for granny time but isn't taking advantage of the opportunity - where she would be responsible.
And yet on the other is suggesting that your dd could live with her.
In a previous post in this thread I mentioned about how my son is up front about what is suitable/convenient and that I go along with the flow. This is an example of his forthrightness.
I have just finished a telephone conversation with him.... I rang him (very long distance international call). I just had a need to hear his voice. (By way of explanation ds and I lived for 22 years with my psychotic schizophrenic DH until I took ds and myself away, before we ended also ended up with health issues from the stress)
It is now 4am here in UK and he is 8 hours behind us i.e. 8pm his time. Ds answered, happy to hear from me, baby under his arm getting baby settled. I asked are you ok to chat, up to your eyes ?? Ds says I'm settling XXXXX but I'm grand for a few minutes. We happily chatted for a few minutes (10). Then Ds said.... Hey Mum, I'm going to have to go now, things to do. I says... no bother.... ...
We both say---- we'll chat/skype soon.
The point I'm making is...... DS speaks up, no offence is meant or taken.
If he speaks up then I know that if I ring and it's not suitable, he will say so. If I'd like a longer chat then it is pre-arranged 
Now.... we are not like the perfect Waltons, we can each irritate the other..... and we know this. If on occasion I have (rarely cos I'm careful) overstepped the mark I am told ---- back off a bit Mum
or, Why would you need to know......
My response is usually.... 'OK, point taken'...... or because I'm a nosey Mum - point still taken".
Veexox if your mil has any sense at all she will conform to a friendly family discussion with regard to establighing ground rules.
So it begins….. Streeting resigns



