Kate,
hmmm... sound like a real war running in your house!! Think my case a bit different as yours. Cos my MIL never help me in taking care Ning. She only "see see look look" when i'm away, and my maid is the one taking care. But, still, she will order the maid to do this and that to Ning. I just pretend "one eye open and one eye close" cos when i'm at home, basically she dun dare to "say" anything. It's becaues, my hb did tell her before, i'm the main taker, so no matter how, she needs to "respect" my way in taking care of my girl. When she gives some opinions, i'll accept if it's ok with me, but if, it's not alright, i'll directly voice out.
So, i can imagine, your MIL is the main taker, and if one day you be SAHM, you wanna take over her job, dun think it's so easy!! Me all the way taking care Ning also will get some tensions with my MIL, not to say you! Anyway, i can say that, our hb plays a very important part also. If your hb can handle your PIL well, then things may change better.
regarding, MIL will not ask their grandson/daughter to "great" the mummy when mummy is back. hahahaha.. i guess it's very very normal for most of the MIL!! They scare that the grandkids will get closer with the DIL than her son or themself.. is a very selfish attitude hor!! My MIL last time the same too!! So, i act from another way, whenever she comes back, i'll ask Ning to "greet" her "po po".. hahahaha.. then slowly, she used to it, when i come back home, she'll ask Ning to call mummy also.
honesty, i believe less DIL 100% willing to stay with PIL.. unless, bo bian, situation need to. So, kate, you need to really discuss with your hb, wanna move out or not.