Hi J. -
I'm sure you feel terribly pitted against the forces here, and I'm sure you'd like to have as much potential chaos as under control as possible when your new baby is born. And I say this with as much loving respect for your feelings as possible -- just because your 2 year old is about to become a big sister, she isn't automatically going to become a big girl, not what you mean by a big girl. A two year old is still a two year old, and she'll have to go through all the developmental stages in her own time. A parent (and grandparent) can discipline, but the expectations simply have to be reasonable.
I highly recommend a book series on child devel - "Your One Year Old, Fun Loving & Fussy," "Your Two Year Old, Terrible & Tender," etc.... Written by Louise Bates Ames and two other authors last names' Ilg and Baker. Excellent reads that help parents understand where their child is inside. They don't lay down dogma, they just explain the developmental stages, saying that some children may be a bit ahead, some may be a bit later. I personally found that understanding what was going on helped me deal. A 2 yr old can be quite tough to live with (so can a 3 yr old, BTW, so get ready, & don't blame your darling daughter when the time comes).
All that said, we as parents do NOT need to be sitting ducks, waiting for intolerable behavior and tantrums to take control of our lives. And grandmas often take short cuts because they are tired at this point in their lives and just don't have the energy for the parenting. But if GM LIVES with you, then she doesn't get to spoil like she would if she was just having a visit.
Perhaps there is some kind of parenting class you could go to together? There's lots of stuff available in the greater LA area. Cedars-Sinai has something they call the Warm Line, where you can call and get parenting advice; they may have a referral. I found a really cool sounding parenting group through Vista Del Mar. I believe the group met in SM. There's a place in Westchester called Emerson that might have something.
Now, this may sound airy-fairy, but I have a final piece of advice, and this only applies if my hunch is correct. Being pregnant, you may be a bit tired and agitated? Perhaps anxious about what's to come? And justifiably frustrated by your MIL's actions which feel like she's undermining you. However --- however... take a moment and look at your actions. Is it possible that you are perhaps acting a bit too angry? If so, your MIL may feel like a bit of softening is in order. She may not be trying to undo your discipline; she may just be trying to comfort. I have no idea, because you haven't said what actually happens in your home when your daughter is in trouble.
Just keep in mind that a 2 yr old barely is aware of why they do what they do. Impulse control is down the road. It's not too soon to teach, but a bit soon to expect a 2 yr old (or even older) to have it right. And, even if she drives you crazy, your MIL will feel more alligned with you if you express appreciation for her help (even if you have to grit your teeth behind your smile & cross your fingers behind your back).
I wish you & your family all the best.
Peace & Blessings,
Colleen