I think there is a difference between Grandma "spoiling" your daughter on her own, and contradicting you in front of her. I think the lack of reaction when your daughter misbehaves is working for you - you walk away and leave her without attention. If you do it without yelling and arguing, she doesn't witness those behaviors and emulate them. That's good! Grandmas ARE for spoiling but not for spoiling your authority. Your daughter should NOT be overhearing your mom say "She's fine" or "Leave her alone."
Maybe you can make an arrangement with your mom to say what she needs to say, but only to you, not within earshot. Maybe she can use other words instead of "It's fine, Mom's not here." Can she say "This is a Grandma Special Treat" or something similar? Not that you want her to go against your rules all the time, but if she changed the language, it's less empowering to your daughter.
ANother thing is to engage your mother in conversations about when you were a child - what she did, how your grandmothers helped out, etc. You might get some insight of how she thought then, what gifts and problems she got from the grandmas or aunts (or even grandpas and uncles), and that might do 2 things: 1) Let you know how she's thinking or what things she is reacting to, swearing to do differently, and 2) give you a possible opening for discussion if it turns out something your grandparents did ticked off your mom! I would do this in the evening after the kids are in bed.
You can also have your mom take over some of the activities - getting ready for school, looking over her homework, etc. Mom may get tired of all the toothbrushing and whatnot.
If none of these halfway measures work, you can get tough and tell her she is leaving you with more problems after she leaves, and setting up your child for failure by endorsing bad behaviors. You can cut back the visits -- the disruption of the schedule is setting off your daughter. She sees an opening, and she's using it. The visits will probably cut back later as your daughter has a bigger homework load, more activities, etc.
What would happen if you used the time with your mom there to go out and do your own errands, do laundry, etc.??? That would remove you from the situation, stop the competition between you and your mom (and I think that's what your momis doing, frankly), and at least stop the contradictions in frontn of you. If Mom gets tired of being the supervisor, she may back off.
Good luck!