If the main reason to for the problem is to reduce childcare costs, then it makes more sense to change one of your days to only have one day off together a week. Both having a Tuesday off and considering still sending her to MIL is a daft idea.
But tbh, I think you're going to regret doing it the way you are doing it. You're already finding her overbearing, she's demanding what she wants, and she's let you down for 50% of the week. Your mum doesn't actually have her by herself at the minute, but you're jumping to 4 days a week of childcare- that's a big amount of time. I'm going to assume both are retired, and hopefully are going to want holidays/ plans with friends/ days out. At the minute, you have no back up plan for that, other than try to take time off, which is not fun in retail!
I personally would look for a nursery/ childminder for at least one day a week. That way, if someone books a holiday or a day away, you can at least ask if they have room for the little one (ours generally could if asked far enough in advance). And if you don't like the childcare set up for whatever reason, if one of the grandparents starts feeding her ice cream for lunch every day when you've said no sugar, or whatever rules you want to follow, - which they will!- you have another possible option there to provide childcare.
Yes childcare costs a fortune. You will qualify for tax free childcare, and some free hours, so it's probably not quite as expensive as you think, but that's the reality of having kids, and it's for 3 years- it's over before you know it.